221A Lecture Notes Path Integral
|
|
- Violet Patrick
- 7 years ago
- Views:
Transcription
1 1A Lecture Notes Path Integral 1 Feynman s Path Integral Formulation Feynman s formulation of quantum mechanics using the so-called path integral is arguably the most elegant. It can be stated in a single line: x f, t f x i, t i Dx(te is[x(t]/ h. (1 The meaning of this equation is the following. If you want to know the quantum mechanical amplitude for a point particle at a position x i at time t i to reach a position x f at time t f, you integrate over all possible paths connecting the points with a weight factor given by the classical action for each path. Hence the name path integral. This is it. Note that the position kets form a complete set of basis, and knowing this amplitude for all x is enough information to tell you everything about the system. The expression is generalized for more dimensions and more particles in a straightforward manner. As we will see, this formulation is completely equivalent to the usual formulation of quantum mechanics. On the other hand, there are many reasons why this expression is just beautiful. First, the classical equation of motion comes out in a very simple way. If you take the limit h 0, the weight factor e is/ h oscillates very rapidly. Therefore, we expect that the main contribution to the path integral comes from paths that make the action stationary. This is nothing but the derivation of Euler Lagrange equation from the classical action. Therefore, the classical trajectory dominates the path integral in the small h limit. Second, we don t know what path the particle has chosen, even when we know what the initial and final positions are. This is a natural generalization of the two-slit experiment. Even if we know where the particle originates from and where it hit on the screen, we don t know which slit the particle came through. The path integral is an infinite-slit experiment. Because you can t specificy where the particle goes through, you sum them up. Third, we gain intuition on what quantum fluctuation does. Around the classical trajectory, a quantum particle explores the vicinity. The trajectory can deviate from the classical trajectory if the difference in the action is 1
2 roughly within h. When a classical particle is confined in a potential well, a quantum particle can go on an excursion and see that there is a world outside the potential barrier. Then it can decide to tunnel through. If a classical particle is sitting at the top of a hill, it doesn t fall; but a quantum particle realizes that the potential energy can go down with a little excursion, and decides to fall. Fourth, whenever we have an integral expression for a quantity, it is often easier to come up with an approximation method to work it out, compared to staring at a differential equation. Good examples are the perturbative expansion and the steepest descent method. One can also think of useful change of variables to simplify the problem. In fact, some techniques in quantum physics couldn t be thought of without the intuition from the path integral. Fifth, the path integral can also be used to calculate partition functions in statistical mechanics. Sixth, the connection between the conservation laws and unitarity transformations become much clearer with the path integral. We will talk about it when we discuss symmetries. There are many more but I stop here. Unfortunately, there is also a downside with the path integral. The actual calculation of a path integral is somewhat technical and awkward. One might even wonder if an integral over paths is mathematically well-defined. Mathematicians figured that it can actually be, but nonetheless it makes some people nervous. Reading off energy eigenvalues is also less transparent than with the Schrödinger equation. Below, we first derive the path integral from the conventional quantum mechanics. Then we show that the path integral can derive the conventional Schrödinger equation back. After that we look at some examples and actual calculations. By the way, the original paper by Feynman on the path integral Rev. Mod. Phys. 0, (1948 is quite readable for you, and I recommend it. Another highly recommended read is Feynman s Nobel Lecture. You will see that Feynman invented the path integral with the hope of replacing quantum field theory with particle quantum mechanics; he failed. But the path integral survived and did mighty good in the way he didn t imagine.
3 Physical Intuition Take the two-slit experiment. Each time an electron hits the screen, there is no way to tell which slit the electron has gone through. After repeating the same experiment many many times, a fringe pattern gradually appears on the screen, proving that there is an interference between two waves, one from one slit, the other from the other. We conclude that we need to sum amplitudes of these two waves that correspond to different paths of the electron. Now imagine that you make more slits. There are now more paths, each of which contributing an amplitude. As you increase the number of slits, eventually the entire obstruction disappears. Yet it is clear that there are many paths that contribute to the final amplitude of the electron propagating to the screen. As we generalize this thought experiment further, we are led to conclude that the amplitude of a particle moving from a point x i to another point x f consists of many components each of which corresponds to a particular path that connects these two points. One such path is a classical trajectory. However, there are infinitely many other paths that are not possible classically, yet contribute to the quantum mechanical amplitude. This argument leads to the notion of a path integral, where you sum over all possible paths connecting the initial and final points to obtain the amplitude. The question then is how you weight individual paths. One point is clear: the weight factor must be chosen such that the classical path is singled out in the limit h 0. The correct choice turns out to be e is[x(t]/ h, where S[x(t] t f t i dtl(x(t, ẋ(t is the classical action for the path x(t that satisfies the boundary condition x(t i x i, x(t f x f. In the limit h 0, the phase factor oscillates so rapidly that nearly all the paths would cancel each other out in the final amplitude. However, there is a path that makes the action stationary, whose contribution is not canceled. This particular path is nothing but the classical trajectory. This way, we see that the classical trajectory dominates the path integral in the h 0 limit. As we increase h, the path becomes fuzzy. The classical trajectory still dominates, but there are other paths close to it whose action is within S h and contribute significantly to the amplitude. The particle does an excursion around the classical trajectory. 3
4 3 Propagator The quantity K(x f, t f ; x i, t i x f, t f x i, t i ( is called a propagator. It knows everything about how a wave function propagates in time, because ψ(x f, t f x f, t f ψ x f, t f x i, t i dx i x i, t i ψ K(x f, t f ; x i, t i ψ(x i, t i dx i. (3 In other words, it is the Green s function for the Schrödinger equation. The propagator can also be written using energy eigenvalues and eigenstates (if the Hamiltonian does not depend on time, K(x f, t f ; x i, t i x f e ih(t f t i / h x i n x f n e ien(t f t i / h n x i n e ien(t f t i / h ψ n(x f ψ n (x i. (4 In particular, Fourier analyzing the propagator tells you all energy eigenvalues, and each Fourier coefficients the wave functions of each energy eigenstates. The propagator is a nice package that contains all dynamical information about a quantum system. 4 Derivation of the Path Integral The basic point is that the propagator for a short interval is given by the classical Lagrangian x 1, t + t x 0, t ce i(l(t t+o( t / h, (5 where c is a normalization constant. This can be shown easily for a simple Hamiltonian H p + V (x. (6 m Note that the expression here is in the Heisenberg picture. The base kets x and the operator x depend on time, while the state ket ψ doesn t. 4
5 The quantity we want is x 1, t + t x 0, t x 1 e ih t/ h x 0 dp x 1 p p e ih t/ h x 0. (7 Because we are interested in the phase factor only at O( t, the last factor can be estimated as p e ih t/ h x 0 p 1 ih t/ h + ( t x 0 ( 1 ī p h m t ī e h V (x 0 t + O( t ipx 0 / h π h 1 exp i ( px 0 + p π h h m t + V (x 0 t + O( t. (8 Then the p-integral is a Fresnel integral x 1, t + t x 0, t dp π h eipx 1/ h e i(px 0+ m πi h t exp ī h p m t+v (x 0 t+o( t / h ( m (x 1 x 0 V (x 0 t + O( t. (9 t The quantity in the parentheses is nothing but the classical Lagrangian times t by identifying ẋ (x 1 x 0 /( t. Once Eq. (5 is shown, we use many time slices to obtain the propagator for a finite time interval. Using the completeness relation many many times, x f, t f x i, t i x f, t f x N1, t N1 dx N1 x N1, t N1 x N, t N dx N dx x, t x 1, t 1 dx 1 x 1, t 1 x i, t i. (10 The time interval for each factor is t (t f t i /N. By taking the limit N, t is small enough that we can use the formula Eq. (5, and we find x f, t f x i, t i N1 i1 dx i e i N1 i0 L(t i t/ h, (11 The expression is asymmetrical between x 1 and x 0, but the difference between V (x 0 t and V (x 1 t is approximately V t(x 0 x 1 V ẋ( t and hence of higher oder. 5
6 up to a normalization. (Here, t 0 t i. In the limit N, the integral over positions at each time slice can be said to be an integral over all possible paths. The exponent becomes a time-integral of the Lagrangian, namely the action for each path. This completes the derivation of the path integral in quantum mechanics. As clear from the derivation, the overall normalization of the path integral is a tricky business. A useful point to notice is that even matrix elements of operators can be written in terms of path integrals. For example, x f, t f x(t 0 x i, t i dx(t 0 x f, t f x(t 0, t 0 x(t 0 x(t 0, t 0 x i, t i dx(t 0 Dx(te is[x(t]/ h x(t 0 Dx(te is[x(t]/ h t f >t>t 0 t 0 >t>t i Dx(te is[x(t]/ h x(t 0, (1 t f >t>t i At the last step, we used the fact that an integral over all paths from x i to x(t 0, all paths from x(t 0 to x f, further integrated over the intermediate position x(t 0 is the same as the integral over all paths from x i to x f. The last expression is literally an expectation value of the position in the form of an integral. If we have multiple insertions, by following the same steps, x f, t f x(t x(t 1 x i, t i Dx(te is[x(t]/ h x(t x(t 1. (13 t f >t>t i Here we assumed that t > t 1 to be consistent with successive insertion of positions in the correct order. Therefore, expectation values in the path integral corresponds to matrix elements of operators with correct ordering in time. Such a product of operators is called timed-ordered T x(t x(t 1 defined by x(t x(t 1 as long as t > t 1, while by x(t 1 x(t if t 1 > t. Another useful point is that Euler Lagrange equation is obtained by the change of variable x(t x(t + δx(t with x i x(t i and x f x(t f held fixed. A change of variable of course does not change the result of the integral, and we find Dx(te is[x(t+δx(t]/ h Dx(te is[x(t]/ h, (14 and hence Dx(te is[x(t+δx(t]/ h Dx(te is[x(t]/ h Dx(te is[x(t]/ h ī δs 0. (15 h 6
7 Recall (see Note on Classical Mechanics I δs S[x(t + δ(t] S[x(t] ( L x d L δx(tdt. (16 dt ẋ Therefore, Dx(te is[x(t]/ h ī h ( L x d L δx(tdt 0. (17 dt ẋ Because δx(t is an arbitrary change of variable, the expression must be zero at all t independently, Dx(te is[x(t]/ h ī h ( L x d L 0. (18 dt ẋ Therefore, the Euler Lagrange equation must hold as an expectation value, nothing but the Ehrenfest s theorem. 5 Schrödinger Equation from Path Integral Here we would like to see that the path integral contains all information we need. In particular, we rederive Schrödinger equation from the path integral. Let us first see that the momentum is given by a derivative. Starting from the path integral x f, t f x i, t i Dx(te is[x(t]/ h, (19 we shift the trajectory x(t by a small amount x(t+δx(t with the boundary condition that x i is held fixed (δx(t i 0 while x f is varied (δx(t f 0. Under this variation, the propagator changes by x f + δx(t f, t f x i, t i x f, t f x i, t i x f x f, t f x i, t i δx(t f. (0 On the other hand, the path integral changes by Dx(te is[x(t+δx(t]/ h Dx(te is[x(t]/ h is[x(t]/ h iδs Dx(te h. (1 7
8 Recall that the action changes by (see Note on Classical Mechanics II δs S[x(t + δx(t] S[x(t] ( tf L L dt δx + t i x ẋ δẋ L ẋ δx x f tf dt x i + t i ( L x d dt L δx. ( ẋ The last terms vanishes because of the equation of motion (which holds as an expectation value, as we saw in the previous section, and we are left with δs L ẋ δx(t f p(t f δx(t f. (3 By putting them together, and dropping δx(t f, we find x f, t f x i, t i x f Dx(te is[x(t]/ h ī h p(t f. (4 This is precisely how the momentum operator is represented in the position space. Now the Schrödinger equation can be derived by taking a variation with respect to t f. Again recall (see Note on Classical Mechanics II S t f H(t f (5 after using the equation of motion. Therefore, is[x(t]/ h i x f, t f x i, t i Dx(te t f h H(t f. (6 If H p + V (x, (7 m the momentum can be rewritten using Eq. (4, and we recover the Schrödinger equation, i h t x f, t f x i, t i 1 ( h + V (x f x f, t f x i, t i. (8 m i x f In other words, the path integral contains the same information as the conventional formulation of the quantum mechanics. 8
9 6 Examples 6.1 Free Particle and Normalization Here we calculate the path integral for a free particle in one-dimension. We need to calculate x f, t f x i, t i Dx(te i m ẋdt/ h, (9 over all paths with the boundary condition x(t i x i, x(t f x f. First, we do it somewhat sloppily, but we can obtain dependences on important parameters nonetheless. We will come back to much more careful calculation with close attention to the overall normalization later on. The classical path is x c (t x i + x f x i t f t i (t t i. (30 We can write x(t x c (t + δx(t, where the left-over piece (a.k.a. quantum fluctuation must vanish at the initial and the final time. Therefore, we can expand δx(t in Fourier series δx(t a n sin nπ (t t i. (31 t f t i The integral over all paths can then be viewed as integrals over all a n, Dx(t c da n. (3 The overall normalization factor c that depends on m and t f t i will be fixed later. The action, on the other hand, can be calculated. The starting point is ẋ ẋ c + δẋ x f x i t f t i + nπ t f t i a n cos πn t f t i (t t i. (33 Because different modes are orthogonal upon t-integral, the action is S m (x f x i t f t i + m 1 (nπ a t f t n. (34 i 9
10 The first term is nothing but the classical action. Then the path integral reduces to an infinite collection of Fresnel integrals, c We now obtain x f, t f x i, t i c da n exp ( [ ( i m (xf x i + h t f t i i π h Therefore, the result is simply m 1 (nπ 1/ exp t f t i x f, t f x i, t i c (t f t i exp [ i h 1 (nπ ] a n. (35 t f t i m [ i h m (x f x i ]. (36 t f t i (x f x i ]. (37 t f t i The normalization constant c can depend only on the time interval t f t i, and is determined by the requirement that dx x f, t f x, t x, t x i, t i x f, t f x i, t i. (38 And hence c (t f tc (t t i πi h(t f t(t t i m(t f t i c (t f t i. (39 We therefore find m c (t πi ht, (40 recovering Eq. (.5.16 of Sakurai precisely. In order to obtain this result directly from the path integral, we should have chosen the normalization of the measure to be m m nπ Dx(t da n. (41 πi h(t f t i πi h(t f t i This argument does not eliminate the possible factor e iω0(t f t i, which would correspond to a zero point in the energy hω 0. Mahiko Suzuki pointed out this problem to me. Here we take the point of view that the normalization is fixed by comparison to the conventional method. 10
11 Now we do it much more carefully with close attention to the overall normalization. The path integral for the time interval t f t i is divided up into N time slices each with t (t f t i /N, where ( K m πi h t S 1 N1 m N/ N1 dx n e is/ h, (4 (x n+1 x n. (43 t The point is that this is nothing but a big Gaussian (to be more precise, Fresnel integral, because the action is quadratic in the integration variables x 1,, x N1. To make this point clear, we rewrite the action into the following form: x N 0 S 1 m t (x N + x 0 m t (x N1, x N, x N3,, x, x m t 0 x x N x N x N3 (x N1, x N, x N3,, x, x x x 1 (44 Comapring to the identity Eq. (106, N dx n e 1 xt Axx T y (π N/ (deta 1/ e + 1 yt A 1 y (45 11
12 we identify A ī m h t 1 1/ / 1 1/ / / / 1 (46 This is nothing but the matrix K N1 in Eq. (106 with a 1/ up to a factor of ī m. One problem is that, for a 1/, λ h t + λ 1/ and hence Eq. (111 is singular. Fortunately, it can be rewritten as detk N1 λn + λ N ( 1 N1 λ N1 + +λ N + λ ++ +λ + λ N +λ N1 N. λ + λ (47 Therefore, ( m N1 deta N 1 ( m N1 i h t N. (48 N1 i h t Therefore, the prefactor of the path integral is ( m N/ (π (N1/ (deta 1/ πi h t ( m N/ ( m (N1/ (π 1/ N 1/ i h t i h t ( ( m 1/ 1/ m. (49 πi hn t πi h(t f t i The exponent is given by x N 0 i 1 im h h t (x N + x ( m (xn, 0, 0,, 0, x 0 A 1 t 0. 0 x 0 (50 We need only (1, 1, (N 1, N 1, (1, N 1, and (N 1, 1 components of A 1. (A 1 1,1 (A 1 N1,N1 i h t detk N i h t N 1 m detk N1 m N. (51 1
13 On the other hand, (A 1 1,N1 (A 1 N1,1 1/ 1 1/ 0 i h t 1 (1 N 0 1/ 1 0 det m detk. N / i h t N1 1 m N N i h t Nm. (5 We hence find the exponent x N 0 i 1 m h t (x N + x ( m (xn, 0, 0,, 0, x 0 A 1 t 0. 0 x 0 ī 1 m h t (x N + x ( im ( i h t N 1 h t m N (x N + x 0 + i h t Nm x Nx 0 ī 1 m 1 h t N (x N + x 0 1 im hn t x Nx 0 ī m (x N x 0 h N t ī m (x f x i. (53 h t f t i Putting the prefactor Eq. (49 and the exponent Eq. (53 together, we find the propagator ( 1/ m K e i m (x f x i h t f t i. (54 πi h(t f t i We recovered Eq. (.5.16 of Sakurai without any handwaving this time! 13
14 6. Harmonic Oscillator Now that we have fixed the normalization of the path integral, we calculate the path integral for a harmonic oscillator in one-dimension. We need to calculate x f, t f x i, t i Dx(te i ( m ẋ 1 mω x dt/ h, (55 over all paths with the boundary condition x(t i x i, x(t f x f. classical path is The sin ω(t f t x c (t x i sin ω(t f t i + x sin ω(t t i f sin ω(t f t i. (56 The action along the classical path is tf ( m S c t i ẋ 1 mω x dt 1 mω (x i + x f cos ω(t f t i x i x f. sin ω(t f t i (57 The quantum fluctuation around the classical path contributes as the O( h correction to the amplitude, relative to the leading piece e isc/ h. We expand the quantum fluctuation in Fourier series as in the case of free particle. Noting that the action is stationary with respect to the variation of x(t around x c (t, there is no linear piece in a n. Because different modes are orthogonal upon t-integral, the action is ( m (nπ 1 S S c + ω (t f t i t f t i a n. (58 Therefore the path integral is an infinite number of Fresnel integrals over a n using the measure in Eq. (41 x f, t f x i, t i e isc/ h m πi h(t f t i [ ( ] m nπ i m (nπ 1 da n exp ω (t f t i πi h(t f t i h t f t i a n ( e isc/ h m 1/ ω(tf t i 1. (59 πi h(t f t i nπ Now we resort to the following infinite product representation of the sine function ( 1 x sin πx n πx. (60 14
15 We find x f, t f x i, t i e isc/ h m ω(t f t i πi h(t f t i sin ω(t f t i e isc/ h mω πi h sin ω(t f t i (61 This agrees with the result from the more conventional method as given in Sakurai Eq. ( Again, we can pay closer attention to the normalization by going through the same steps as for a free particle using discretized time slices. The action is S 1 N1 m (x n+1 x n t 1 N1 mω x n 1 mω ( 1 x x N. (6 The last term is there to correctly acccount for all N 1 time slices for the potential energy term V dt, but to retain the symmetry between the initial and final positions, we took their average. Once again, this is nothing but a big Fresnel (complex Gaussian integral: x N 0 S 1 m t (x N + x 0 m t (x N1, x N, x N3,, x, x m t 0 x 0 (x N1, x N, x N3,, x, x 1 1 mω (x N1, x N, x N3,, x, x x N1 x N x N3 15. x x 1 x N1 x N x N3. x x 1 t 1 4 mω (x N + x 0 t
16 (63 Comapring to the identity Eq. (106, N dx n e 1 xt Axx T y (π N/ (deta 1/ e + 1 yt A 1 y (64 we identify A ī h mω ti N1 ī m h t 1 1/ / 1 1/ / / / 1 (65 This is nothing but the matrix K N1 in Eq. (106 up to an overall factor of with Using Eq. (111, ī m h t + ī h mω t a 1 λ ± 1 m i h t ( ω ( t (66 m/ t m t mω t 1 ω ( t. (67 ( 1 ± 1 4 ( ω ( t (68 Therefore, deta ( m N1 ( ω ( t N1 λn + λ N. (69 i h t λ + λ Therefore, the prefactor of the path integral is ( m N/ (π (N1/ (deta 1/ πi h t ( m ( ω ( t (N1/ λ N + λ N 1/. (70 πi h t λ + λ 16
17 Now we use the identity ( lim 1 + x N e x. (71 N N We find in the limit of t (t f t i /N and N, λ ± 1 ( 4 1 ± 1 1 ( ω ( t (1 ± iω t + O(N 3, (7 and hence In the same limit, λ N + λ N 1 e iω(tf ti e iω(tf ti + O(N 1 λ + λ N iω t N sin ω(t f t i + O(N 1. (73 N1 ω(t f t i ( ω ( t (N1/ Therefore, the prefactor Eq. (70 is m πi h t (N1/ The exponent is given by ( N sin ω(t f t i N1 ω(t f t i i 1 im h h t (x N+x 0 ī 1 h 4 mω (x N+x 0+ 1 (N1/ ( 1 1 ω ( t (N1/ (N1/ + O(N 1. (74 1/ ( im h t mω πi h sin ω(t f t i. (75 x N 0 (x N, 0, 0,, 0, x 0 A (76 We need only (1, 1, (N 1, N 1, (1, N 1, and (N 1, 1 components of A 1. (A 1 1,1 (A 1 N1,N1 i h t m ( ω ( t 1 detk N i h t detk N1 m ( ω ( t 1 λn1 + λ N1 λ N + λ N (77 17 x 0
18 On the other hand, (A 1 1,N1 (A 1 N1,1 a 1 a 0 i h t m ( ω ( t 1 1 (1 N 0 a 1 0 det detk. N a i h t m ( ω ( t 1 λ + λ a N λ N + λ N i h t N1 ω(t f t i 1 i h t ω(t f t i 1 m N sin ω(t f t i N Nm sin ω(t f t i. (78 We hence find the exponent x N 0 ( i 1 im h h t 1 4 mω (x N + x ( m (xn, 0, 0,, 0, x 0 A 1 t 0. 0 x 0 ī h ( ( 1 im h t 1 4 mω ( ω ( t 1 λn1 (x N + x ( im i h t h t m + λ N1 (x λ N + λ N N + x 0 + N ω(t f t i sin ω(t f t i x Nx 0 (79 Taking the limit N, the second term is obviously regular, and the last term in the parentheses is also: ( 1 im i h t ω(t f t i h t m N sin ω(t f t i x Nx 0 ī mω h sin ω(t f t i x Nx 0. (80 The other terms are singular and we need to treat them carefully. i 1 m h t (x N + x 0 1 im h t ( ω ( t 1 λn1 + λ N1 (x λ N + λ N N + x 0 im ( 1 ( ω ( t 1 λn1 + λ N1 (x h t λ N + λ N N + x 0. (81 18
19 We need O(N 1 term in the parentheses. The factor ( ω ( t O(N is easy. Using Eq. (7, the next factor is λ N1 + λ N1 λ N + λ N N (1 + iω t N1 (1 iω t N1 + O(N (8 N1 (1 + iω t N (1 iω t N + O(N Therefore, dropping O(N corrections consistently, 1 ( ω ( t 1 λn1 + λ N1 λ N + λ N (1 + iω tn (1 iω t N (1 + iω t N1 + (1 iω t N1 (1 + iω t N (1 iω t N iω t (1 + iω tn1 + (1 iω t N1 (1 + iω t N (1 iω t N iω (t f t i N cos ω(t f t i i sin ω(t f t i 1 N Collecting all the terms, the exponent is i h S c ī mω h sin ω(t f t i x Nx 0 + im 1 h t N ī mω (x N + x 0 cos ω(t f t i x N x 0 h sin ω(t f t i which is the same as the classical action Eq. (57. 7 Partition Function ω(t f t i cos ω(t f t i. (83 sin ω(t f t i ω(t f t i cos ω(t f t i (x N + x sin ω(t f t i 0 (84 The path integral is useful also in statistical mechanics to calculate partition functions. Starting from a conventional definition of a partition function Z n e βen, (85 where β 1/(k B T, we rewrite it in the following way using completeness relations in both energy eigenstates and position eigenstates. Z n e βh n dx n x x e βh n n n dx x e βh n n x dx x e βh x. (86 n 19
20 The operator e βh is the same as e iht/ h except the analytic continuation t iτ i hβ. Therefore, the partition function can be written as a path integral for all closed paths, i.e., paths with the same beginning and end points, over a time interval i hβ. For a single particle in the potential V (x, it is then Z Dx(τ exp 1 h hβ 0 dτ m ( x τ + V (x. (87 The position satisfies the periodic boundary condition x( hβ x(0. For example, in the case of a harmonic oscillator, we can use the result in the previous section x f, t f x i, t i [ mω i πi h sin ω(t f t i exp 1 h mω (x i + x ] f cos ω(t f t i x i x f, sin ω(t f t i and take x f x i x, t f t i i hβ (88 [ x e βh mω x π h sinh β hω exp 1 h ] 1 cosh β hω mωx. (89 sinh β hω Then integrate over x, Z dx x e βh x mω π h sinh β hω 1 4 sinh β hω/ 1 sinh β hω/ eβ hω/ π h mω sinh β hω cosh β hω 1 e β hω e β(n+ 1 hω. (90 n0 0
21 Of course, we could have redone the path integral to obtain the same result. The only difference is that the classical path you expand the action around is now given by sinh rather than sin. Let us apply the path-integral representation of the partition function to a simple harmonic oscillator. The expression is Z Dx(τ exp 1 h hβ 0 dτ m ( x + m τ ω x We write all possible paths x(τ in Fourier series as x(τ a 0 + a n cos πn hβ τ +. (91 b n sin πn τ, (9 hβ satisfying the periodic boundary condition x( hβ x(0. Then the exponent of the path integral becomes 1 hβ dτ m ( x + m h 0 τ ω x βm a ( πn + ω (a n + b hβ n. (93 The integration over all possible paths can be done by integrating over the Fourier coefficients a n and b n. Therefore the partition function is Z c da 0 da n db n exp βm ω a ( πn + ω (a n + b hβ n. (94 This is an infinite collection of Gaussian integrals and becomes ( π 1 4π Z c πn + ω. (95 βmω βm hβ Here, c is a normalization constant which can depend on β because of normalization of Fourier modes, but not on Lagrangian parameters such as m or ω. Now we use the infinite product representation of the hyperbolic function ( 1 + x n 1 sinh πx πx. (96
22 The partition function Eq. (95 can be rewritten as π Z c βmω c sinh hβω/ h β 1 + πmn ( hβω πn 1 c e hβω/, (97 1 e hβω where c is an overall constant, which does not depend on ω, is not important when evaluating various thermally averages quantities. If you drop c, this is exactly the partition function for the harmonic oscillator, including the zero-point energy. A Path Integrals in Phase Space As we saw above, the normalization of the path integral is a somewhat tricky issue. The source of the problem was when we did p integral in Eq. (9, that produced a singular prefactor m/πi h t. On the other hand, if we do not do the p integral, and stop at dp x 1, t + t x 0, t eipx1/ h e i(px0+ p m t+v (x0 t+o( t / h π h dp x 1 x 0 π h ei(p t p m V (x0 t, (98 we do not obtain any singular prefactor except 1/π h for every momentum integral. The path integral can then be written as x f, t f x i, t i Dx(tDp(te is[x(t,p(t]/ h, (99 where the action is given in the phase space S[x(t, p(t] tf t i (pẋ H(x, p dt. (100 The expression (99 is an integral over all paths in the phase space (x, p. It is understood that there is one more p integral than the x integral (albeit both infinite because of the number of times the completeness relations are inserted. Using the path integral in the phase space, we can work out the precise normalization of path integrals up to numerical constants that depend on t, but not on m, or any other
23 parameters in the Lagrangian. For instance, for the harmonic oscillator, we expand x(t x c (t + x n sin nπ (t t i, (101 t f t i t f t i 1 nπ p(t p c (t + p 0 + p n cos (t t i. (10 t f t i t f t i t f t i The action is then S S c p 0 m 1 (p n, x n ( 1 m nπ t f t i nπ t f t i mω ( pn, (103 x n where S c is the action for the classical solution Eq. (57. Using the integration volume we find the path integral to be Dx(tDp(t c(t f t i dp0 π h dx n dp n π h, (104 x f, t f x i, t i e isc/ h c(t f t i 1 ( 1/ πm h π h i ( 1/ ( 1/ 1 πm h π h 1 π h i im ω (nπ/(t f t i. (105 Up to an overall constant c(t f t i that depends only on t f t i, both m and ω dependences are obtained correctly. B A Few Useful Identities To carry out the integral, a few identities would be useful. For N 1-dimensional column vectors x, y, and a symmetric matrix A, N dx n e 1 xt Axx T y (π N/ (deta 1/ e + 1 yt A 1 y (106 It is easy to prove this equation first without the linear term y by diagonalizatin the matrix A O T DO, where O is a rotation matrix and D diag(λ 1,, λ N has N eigenvalues. Under a rotation, the measure dx n is invariant because the Jacobian is deto 1. Therefore, after rotation of the integration variables, the integral is N dx n e λnx n / N π N (π N/ ( λ n 1/ (107 λ n 3
24 The product of eigenvalues is nothing but the determinant of the matrix deta deto T detddeto N λ n, and hence N dx n e 1 xt Ax (π N/ (deta 1/. (108 In the presence of the linear term, all we need to do is to complete the square in the exponent, 1 xt Ax x T y 1 (xt y T A 1 A(x A 1 y + 1 yt A 1 y, (109 and shift x to eliminate A 1 y, and diagonalize A. This proves the identity Eq. (106. For a N N matrix of the form 1 a a 1 a a K N......, ( a a 1 one can show detk N λn+1 + λ N+1, λ ± 1 λ + λ (1 ± 1 4a. (111 This is done by focusing on the top two-by-two block. The determinant that picks the (1, 1 element 1 comes with the determinant of remaining (N 1 (N 1 block, namely detk N1. If you pick the (1, element a, it is necessary to also pick the (, 1 element which is also a and the rest comes from the remaining (N (N block, namely detk N. Therefore, we find a recursion relation We rewrite this recursion relation as ( ( detkn 1 a detk N1 1 0 detk N detk N1 a detk N. (11 ( detkn1. (113 detk N The initial conditions are detk 1 1, detk 1 a. Using the recursion relation backwards, it is useful to define detk 0 1. We diagonalize the matrix ( ( ( ( 1 a λ+ λ λ+ 0 1 λ 1. ( λ 1 λ + λ + λ Hence, ( detkn detk N1 ( λ+ λ 1 1 ( N1 ( λ+ 0 1 λ 0 λ 1 λ + 1 λ + λ ( detk1 detk 0 4
25 ( ( 1 λ+ λ λ N1 + 0 λ + λ λ N1 ( ( 1 λ N + λ N λ+ λ + λ λ+ N1 λ N1 λ ( 1 λ N+1 + λ N+1 λ + λ λ N + λ N ( 1 λ 1 + λ +. (115 5
1 Variational calculation of a 1D bound state
TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK FÖRDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN- ICS Kvantmekanik fördjupningskurs SI38 för F4 Thursday December, 7, 8. 13. Write on each page: Name,
More informationPath Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demonstrate
More informationLet s first see how precession works in quantitative detail. The system is illustrated below: ...
lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationElasticity Theory Basics
G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationIntroduction to Schrödinger Equation: Harmonic Potential
Introduction to Schrödinger Equation: Harmonic Potential Chia-Chun Chou May 2, 2006 Introduction to Schrödinger Equation: Harmonic Potential Time-Dependent Schrödinger Equation For a nonrelativistic particle
More informationTeoretisk Fysik KTH. Advanced QM (SI2380), test questions 1
Teoretisk Fysik KTH Advanced QM (SI238), test questions NOTE THAT I TYPED THIS IN A HURRY AND TYPOS ARE POSSIBLE: PLEASE LET ME KNOW BY EMAIL IF YOU FIND ANY (I will try to correct typos asap - if you
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationTill now, almost all attention has been focussed on discussing the state of a quantum system.
Chapter 13 Observables and Measurements in Quantum Mechanics Till now, almost all attention has been focussed on discussing the state of a quantum system. As we have seen, this is most succinctly done
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More information8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Problem Set 5
8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Tuesday March 5 Problem Set 5 Due Tuesday March 12 at 11.00AM Assigned Reading: E&R 6 9, App-I Li. 7 1 4 Ga. 4 7, 6 1,2
More informationQuantum Mechanics and Representation Theory
Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30
More information2.2 Magic with complex exponentials
2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationLecture L3 - Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationarxiv:0805.1170v1 [cond-mat.stat-mech] 8 May 2008
Path integral formulation of fractional Brownian motion for general Hurst exponent arxiv:85.117v1 [cond-mat.stat-mech] 8 May 28 I Calvo 1 and R Sánchez 2 1 Laboratorio Nacional de Fusión, Asociación EURAOM-CIEMA,
More informationLectures 5-6: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationOperator methods in quantum mechanics
Chapter 3 Operator methods in quantum mechanics While the wave mechanical formulation has proved successful in describing the quantum mechanics of bound and unbound particles, some properties can not be
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationHow do we obtain the solution, if we are given F (t)? First we note that suppose someone did give us one solution of this equation
1 Green s functions The harmonic oscillator equation is This has the solution mẍ + kx = 0 (1) x = A sin(ωt) + B cos(ωt), ω = k m where A, B are arbitrary constants reflecting the fact that we have two
More informationOn a class of Hill s equations having explicit solutions. E-mail: m.bartuccelli@surrey.ac.uk, j.wright@surrey.ac.uk, gentile@mat.uniroma3.
On a class of Hill s equations having explicit solutions Michele Bartuccelli 1 Guido Gentile 2 James A. Wright 1 1 Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK. 2 Dipartimento
More informationarxiv:1603.01211v1 [quant-ph] 3 Mar 2016
Classical and Quantum Mechanical Motion in Magnetic Fields J. Franklin and K. Cole Newton Department of Physics, Reed College, Portland, Oregon 970, USA Abstract We study the motion of a particle in a
More informationThe one dimensional heat equation: Neumann and Robin boundary conditions
The one dimensional heat equation: Neumann and Robin boundary conditions Ryan C. Trinity University Partial Differential Equations February 28, 2012 with Neumann boundary conditions Our goal is to solve:
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More information2.6 The driven oscillator
2.6. THE DRIVEN OSCILLATOR 131 2.6 The driven oscillator We would like to understand what happens when we apply forces to the harmonic oscillator. That is, we want to solve the equation M d2 x(t) 2 + γ
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationFourier Analysis. u m, a n u n = am um, u m
Fourier Analysis Fourier series allow you to expand a function on a finite interval as an infinite series of trigonometric functions. What if the interval is infinite? That s the subject of this chapter.
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationNotes on Quantum Mechanics
Notes on Quantum Mechanics K. Schulten Department of Physics and Beckman Institute University of Illinois at Urbana Champaign 405 N. Mathews Street, Urbana, IL 680 USA April 8, 000 Preface i Preface The
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1
19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point
More informationSecond Order Linear Partial Differential Equations. Part I
Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; - point boundary value problems; Eigenvalues and Eigenfunctions Introduction
More informationChapter 22 The Hamiltonian and Lagrangian densities. from my book: Understanding Relativistic Quantum Field Theory. Hans de Vries
Chapter 22 The Hamiltonian and Lagrangian densities from my book: Understanding Relativistic Quantum Field Theory Hans de Vries January 2, 2009 2 Chapter Contents 22 The Hamiltonian and Lagrangian densities
More informationTime dependence in quantum mechanics Notes on Quantum Mechanics
Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:37-05:00 Copyright 2003 Dan
More information8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
More informationLecture 1: Schur s Unitary Triangularization Theorem
Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections
More informationQuantum Mechanics: Postulates
Quantum Mechanics: Postulates 5th April 2010 I. Physical meaning of the Wavefunction Postulate 1: The wavefunction attempts to describe a quantum mechanical entity (photon, electron, x-ray, etc.) through
More informationIntroduction to Complex Numbers in Physics/Engineering
Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The
More information5 Homogeneous systems
5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m
More information4 Lyapunov Stability Theory
4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We
More information5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
More informationSeminar 4: CHARGED PARTICLE IN ELECTROMAGNETIC FIELD. q j
Seminar 4: CHARGED PARTICLE IN ELECTROMAGNETIC FIELD Introduction Let take Lagrange s equations in the form that follows from D Alembert s principle, ) d T T = Q j, 1) dt q j q j suppose that the generalized
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004
PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall
More informationN 1. (q k+1 q k ) 2 + α 3. k=0
Teoretisk Fysik Hand-in problem B, SI1142, Spring 2010 In 1955 Fermi, Pasta and Ulam 1 numerically studied a simple model for a one dimensional chain of non-linear oscillators to see how the energy distribution
More informationLecture 11. 3.3.2 Cost functional
Lecture 11 3.3.2 Cost functional Consider functions L(t, x, u) and K(t, x). Here L : R R n R m R, K : R R n R sufficiently regular. (To be consistent with calculus of variations we would have to take L(t,
More informationThree Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009
Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce
More informationThe 1-D Wave Equation
The -D Wave Equation 8.303 Linear Partial Differential Equations Matthew J. Hancock Fall 006 -D Wave Equation : Physical derivation Reference: Guenther & Lee., Myint-U & Debnath.-.4 [Oct. 3, 006] We consider
More informationFLAP P11.2 The quantum harmonic oscillator
F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P. Opening items. Module introduction. Fast track questions.3 Ready to study? The harmonic oscillator. Classical description of
More informationThe continuous and discrete Fourier transforms
FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationMASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)
MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for non-physics majors, a GPA of 2.5 or better in at least 15 units in the following advanced
More information(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7)
Chapter 4. Lagrangian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7 4.1 Important Notes on Notation In this chapter, unless otherwise stated, the following
More informationScientific Programming
1 The wave equation Scientific Programming Wave Equation The wave equation describes how waves propagate: light waves, sound waves, oscillating strings, wave in a pond,... Suppose that the function h(x,t)
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationCHAPTER IV - BROWNIAN MOTION
CHAPTER IV - BROWNIAN MOTION JOSEPH G. CONLON 1. Construction of Brownian Motion There are two ways in which the idea of a Markov chain on a discrete state space can be generalized: (1) The discrete time
More informationTime Ordered Perturbation Theory
Michael Dine Department of Physics University of California, Santa Cruz October 2013 Quantization of the Free Electromagnetic Field We have so far quantized the free scalar field and the free Dirac field.
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
More informationCHAPTER 2. Eigenvalue Problems (EVP s) for ODE s
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS
More informationApplication of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semi-infinite domain)
Application of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semi-infinite domain) The Fourier Sine Transform pair are F. T. : U = 2/ u x sin x dx, denoted as U
More informationDifferentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
More informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More informationarxiv:1201.6059v2 [physics.class-ph] 27 Aug 2012
Green s functions for Neumann boundary conditions Jerrold Franklin Department of Physics, Temple University, Philadelphia, PA 19122-682 arxiv:121.659v2 [physics.class-ph] 27 Aug 212 (Dated: August 28,
More information1 Lecture 3: Operators in Quantum Mechanics
1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators
More informationAssessment Plan for Learning Outcomes for BA/BS in Physics
Department of Physics and Astronomy Goals and Learning Outcomes 1. Students know basic physics principles [BS, BA, MS] 1.1 Students can demonstrate an understanding of Newton s laws 1.2 Students can demonstrate
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationAn Introduction to Hartree-Fock Molecular Orbital Theory
An Introduction to Hartree-Fock Molecular Orbital Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2000 1 Introduction Hartree-Fock theory is fundamental
More informationENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE ORDINARY DIFFERENTIAL EQUATIONS
Texas College Mathematics Journal Volume 6, Number 2, Pages 18 24 S applied for(xx)0000-0 Article electronically published on September 23, 2009 ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE
More informationChapter 2. Parameterized Curves in R 3
Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationFernanda Ostermann Institute of Physics, Department of Physics, Federal University of Rio Grande do Sul, Brazil. fernanda.ostermann@ufrgs.
Teaching the Postulates of Quantum Mechanics in High School: A Conceptual Approach Based on the Use of a Virtual Mach-Zehnder Interferometer Alexsandro Pereira de Pereira Post-Graduate Program in Physics
More informationHøgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point
More informationThe Method of Least Squares. Lectures INF2320 p. 1/80
The Method of Least Squares Lectures INF2320 p. 1/80 Lectures INF2320 p. 2/80 The method of least squares We study the following problem: Given n points (t i,y i ) for i = 1,...,n in the (t,y)-plane. How
More informationEuler s Method and Functions
Chapter 3 Euler s Method and Functions The simplest method for approximately solving a differential equation is Euler s method. One starts with a particular initial value problem of the form dx dt = f(t,
More informationContinuous Groups, Lie Groups, and Lie Algebras
Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups
More informationSolutions to Linear First Order ODE s
First Order Linear Equations In the previous session we learned that a first order linear inhomogeneous ODE for the unknown function x = x(t), has the standard form x + p(t)x = q(t) () (To be precise we
More information15. Symmetric polynomials
15. Symmetric polynomials 15.1 The theorem 15.2 First examples 15.3 A variant: discriminants 1. The theorem Let S n be the group of permutations of {1,, n}, also called the symmetric group on n things.
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
More informationThe two dimensional heat equation
The two dimensional heat equation Ryan C. Trinity University Partial Differential Equations March 6, 2012 Physical motivation Consider a thin rectangular plate made of some thermally conductive material.
More informationCOMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS
COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS BORIS HASSELBLATT CONTENTS. Introduction. Why complex numbers were introduced 3. Complex numbers, Euler s formula 3 4. Homogeneous differential equations 8 5.
More information5.4 The Heat Equation and Convection-Diffusion
5.4. THE HEAT EQUATION AND CONVECTION-DIFFUSION c 6 Gilbert Strang 5.4 The Heat Equation and Convection-Diffusion The wave equation conserves energy. The heat equation u t = u xx dissipates energy. The
More informationDerive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
More informationSecond Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.
More informationHow Gravitational Forces arise from Curvature
How Gravitational Forces arise from Curvature 1. Introduction: Extremal ging and the Equivalence Principle These notes supplement Chapter 3 of EBH (Exploring Black Holes by Taylor and Wheeler). They elaborate
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More information