TOOLS FOR THE QUANTUM MECHANIC

Transcription

1 TOOLS FOR THE QUANTUM MECHANIC JEFF MARSH It became apparent in the early years of the twentieth century that the framework physicists had built for understanding the material universe, which had worked so well for over 200 years, was inadequate for describing phenomena on the atomic scale. The development of a new theory, which came to be known as quantum mechanics, constituted nothing less than a revolution in Physics. (For an entertaining account of these developments, written by one of the participants, see [8]; a less entertaining but more mathematical narrative may be found in [4].) The new theory required new mathematics, much of which was invented (or re-invented) ad hoc without regard to formal definition or mathematical rigor. Like the clean-up after a particularly successful office party, it was left for others to formalize the mathematics. In this paper I recount the highlights of the three-decade party that gave rise to quantum mechanics, and the subsequent mathematical clean-up. 1. A Brief History of Quantum Mechanics Quantum mechanics developed along two important but initially separate paths. The first was the realization that energy is exchanged between atomic systems in discrete amounts called quanta. (The Latin quantum means how much? ) The second was the discovery that light has both wave-like and particle-like properties, the so-called wave-particle duality of light. In the year 1900 Max Planck, in order to explain the spectrum of thermal radiation, hypothesized that matter emits light energy in small, indivisible units, which he called energy quanta [17, 18] (for English-language translations see [26]). This energy is proportional to the frequency of the light, E = hν, where E is the energy, and ν is the frequency. The constant of proportionality, h, now known as Planck s constant, is a very small number, h = Joule-seconds. The energy/frequency relation is usually written E = ω, where ω = 2πν is the radian frequency, and = h/2π (pronounced h bar ) is also referred to as Planck s constant. Planck s theory was a complete break with the Physics of the time which held that light energy could be emitted in a continuous range. Max Planck was awarded the 1918 Nobel prize in Physics for this work. Albert Einstein in 1905 published a paper that explained a phenomena known as the photoelectric effect by assuming that matter absorbs light Date: 2008 May 8. Submitted in partial satisfaction of the requirements of the course Mathematics 535: Functional Analysis. 1

2 2 JEFF MARSH energy in small, indivisible units [7] (for an English-language translation see [1]). Einstein is remembered today primarily for the special theory of relativity, however it was his work on the photoelectric effect for which he was awarded the Nobel prize in On a macroscopic scale light surely behaves like a wave. However, if on a subatomic scale light is both emitted and absorbed in discrete units, it makes sense to describe light as a particle. Thus was the wave-particle duality of light established. The word photon was coined in 1926 (by a chemist!) [14] as a term for a particle of light. In what must be the craziest theory ever proposed in the history of Physics (craziest theory that actually turned out to be correct, that is), Louis de Broglie in 1924 proposed that the wave-particle duality of light extended to matter as well [5] (for an English-language translation see [15]). Every particle of matter has an associated matter wave which obeys not only Planck s E = ω, but also p = h/λ, where p is the particle s momentum and λ is the wavelength of the wave. (This relation is usually written as p = k where k = 2π/λ is known as the wave number or sometimes the propagation constant.) This theory was de Broglie s PhD thesis, for which he was awarded the Nobel prize in Pause to consider the two equations: (1) (2) E = ω p = k Each has a particle-like property (energy, momentum) on the left and a wavelike property (frequency, wave number) on the right, with the ubiquitous Planck s constant, Janus-like, in-between. In 1926, following a seminar about the de Broglie hypothesis, Peter Debye remarked to a fellow physicist that if matter is a wave there should be a matter-wave equation [9, p. 10]. Later that year the physicist to which the remark was made, Erwin Schrödinger, published the differential wave equation for matter now known as the Schrödinger equation [21, 22, 23, 24] (for English-language translations see [15]). Schrödinger shared the 1933 Nobel prize in Physics with Paul A. M. Dirac. Schrödinger s formulation of quantum mechanics was not the first, however. In 1925 Werner Heisenberg announced a quantum theory that was to become known as matrix mechanics [10, 3, 2] (for English-language translations see [27]). In this theory the dynamical variables (such as position and momentum) that appear in the classical equations of motion are replaced by infinite-dimensional matrices. Heisenberg received the Nobel prize in The Schrödinger and Heisenberg theories are conceptually and mathematically completely different. Schrödinger s theory involves continuous functions which are the solutions of a differential equation. Heisenberg s theory involves discrete quantities obtained from algebraic equations. Yet the two theories give the same results. This state of affairs, having two very different theories producing the same physical results, was the impetus that led to

3 TOOLS FOR THE QUANTUM MECHANIC 3 a deeper understanding of the mathematics involved, and it was eventually established that the two theories are mathematically equivalent. The credit for putting quantum mechanics on a firm mathematical foundation goes to John von Neumann [28], who never won a Nobel prize. 2. The Classical Harmonic Oscillator We will illustrate the formalism and methodology of quantum mechanics in the context of a simple model, the harmonic oscillator. The harmonic oscillator is a very nice model; it is exactly solvable in both the classical and quantum cases, and may be used to describe the behavior of a system about any stable equilibrium for sufficiently small oscillations. Even in its onedimensional form it exhibits qualitative features of much more complicated physical systems, such as atoms and molecules. Consider a particle of mass m moving under the influence of a timeindependent potential energy function V (x). The state of this system at any time is uniquely specified by the particle s position x, and its momentum p which is defined as p = m dt The total energy E of the particle is constant and is given by (3) p 2 2m + V (x) = E The harmonic oscillator potential is V (x) = 1 2 κx2 where κ is a positive constant. It is not difficult to solve the equation of motion in this case to obtain x(t) = A cos(ω o t + δ) where A, ω o, and δ are constants. The constant ω o is the natural frequency of oscillation of the particle and is given by ω o = κ/m. This relation allows us to rewrite the potential as V (x) = 1 2 mω2 ox 2 and we henceforth do so. The constant δ is an arbitrary phase, and the amplitude A can be related to the total energy as A = 1 ω 0 2E m The particle is strictly confined to be within a distance A from the origin which is a function of the total energy. Note that there is no restriction on E other than it be non-negative. 3. The Schrödinger Equation The Schrödinger equation in one spatial dimension, for a particle of mass m with no internal degrees of freedom and moving in a time-independent potential V (x) is 2 (4) 2 2m x 2 Ψ(x, t) + V (x)ψ(x, t) = i Ψ(x, t) t

4 4 JEFF MARSH The complex-valued function Ψ(x, t) is the sought-for matter wave associated with the particle and is consistent with the de Broglie hypothesis as follows: For a free particle (V (x) = 0) a possible solution to (4) is the harmonic de Broglie wave Ψ(x, t) = e i(kx ωt) Using the relations (1) and (2) we write this as Ψ(x, t) = e i(px Et)/ Substituting into (4) we obtain p 2 /2m = E, which is the correct (nonrelativistic) relation between momentum and energy for a free particle. We must now relate properties of the wave function Ψ(x, t) to the behavior of the associated particle. The probability density P (x, t) is defined such that P (x, t) is the probability that a position measurement at time t will find the particle between x and x +. The Born postulate states that P (x, t) = Ψ (x, t)ψ(x, t) where Ψ (x, t) is the complex conjugate of Ψ(x, t). Note that this relation guarantees that P (x, t) is real. Also, the interpretation of Ψ (x, t)ψ(x, t) as a probability density requires that Ψ(x, t) be normalized such that (5) + Ψ (x, t)ψ(x, t) = 1 Insofar as it is the probability density that is the actual physical reality, the wave function Ψ(x, t) is arbitrary up to a constant phase e iθ. If we write Ψ(x, t) = ψ(x)φ(t), equation (4) yields to the usual procedure of separation of variables. The time-independent Schrödinger equation is then (6) 2 d 2 ψ(x) 2m 2 + V (x)ψ(x) = Eψ(x) where we have written the separation constant as E, which turns out to be identified with the total energy of the particle. The time dependence of the wave function is very interesting, but will not concern us here. Of course, the wave functions ψ(x) must also satisfy the normalization condition (5). To be normalizable, the integral of ψ (x)ψ(x) = ψ(x) 2 over all x must be finite and we say ψ(x) is square-integrable. The space of all such functions is designated as L 2 (, + ) and we write ψ L 2 (, + ). 4. The Quantum Harmonic Oscillator The time-independent Schrödinger equation (6) for the harmonic oscillator potential becomes 2 d 2 ψ(x) 2m mω2 ox 2 ψ(x) = Eψ(x)

5 TOOLS FOR THE QUANTUM MECHANIC 5 By introducing the dimensionless variable u = x mω o / we can re-write this as d 2 ψ(u) du 2 + (ɛ u 2 )ψ(u) = 0 where ɛ is the dimensionless quantity 2E/ ω o. The requirement that ψ L 2 (, + ) (which can be considered a boundary condition) places severe restrictions on the possible solutions, and we find that such solutions exist only for specific values of ɛ given by ɛ n = 2n + 1, n = 0, 1, 2,... which correspond to the particle s energy as E n = 1 2 ɛ n ω o = (n ) ω o, n = 0, 1, 2,... Thus we see that the energy values allowed in a quantum oscillator are quantized. Energy quantization is naturally described by solutions of the Schrödinger equation. 1 The difference between adjacent energy values is E = E n+1 E n = ω o. Since the oscillator can exist only in states corresponding to integer n, a transition between states must be accompanied by absorption or emission of energy equal to an integer multiple of ω o. This result is nicely consistent with the quantum hypothesis of Planck and Einstein. The solution corresponding to n = 0 is called the ground state of the oscillator, and we see that the energy of the ground state is not zero but 1 2 ω o. This is called the zero-point energy. The zero-point energy is responsible for some interesting phenomena seen in matter at low temperatures, such as the fact that liquid helium will not freeze into a solid even at a temperature of absolute zero. For each n we have the solution ψ n (u) = C n e u2 /2 H n (u) where C n is the normalization constant C n = 2 n/2 π 1/4 (n!) 1/2 and the H n (u) are a complete set of orthogonal functions known as the Hermite polynomials. Figure 1 shows the first few wave functions ψ n (u). Note that for n = 0, 2, 4,... the functions are even (i.e., ψ(u) = ψ( u)), and for n = 1, 3, 5,... the functions are odd (i.e., ψ(u) = ψ( u)). We say the even functions have parity +1 and the odd functions have parity 1. 1 The energies associated with the harmonic oscillator solutions are quantized, however this is not always the case for a given potential function V (x).

6 6 JEFF MARSH n=0 n=1 n=2 n=3 n=4 n=5 Figure 1. The first six quantum harmonic oscillator wave functions ψ n (u). The dotted lines indicate the limits of motion for the classical harmonic oscillator. 5. The Schrödinger Equation as an Operator Equation The determination of the properties of the quantum harmonic oscillator in the previous section was essentially an exercise in solving a differential equation. Lest one think quantum mechanics is nothing more than solving differential equations, we now present a slightly more abstract interpretation of the Schrödinger equation. This abstraction is motivated by the idea that the wave function ψ(x) completely represents the state of a quantum mechanical system; everything that can be known about the particle in question is somehow encoded in its wave function. Our task now is to crack the code. We rewrite the time-independent Schrödinger equation (6) as ( 1 i d ) 2 ψ(x) + V (x)ψ(x) = Eψ(x) 2m Comparison with equation (3) suggests we define a differential operator for the momentum ˆp = i d

7 TOOLS FOR THE QUANTUM MECHANIC 7 where we use the hat notation to indicate that ˆp is an operator. We now write the Schrödinger equation as ( ) ˆp 2 2m + V (x) ψ(x) = Eψ(x) In some funny sense, ˆp operates on the wave function and extracts the momentum information from it, which we then use to calculate the total energy. Viewed in this light, the Schrödinger equation appears to be a statement of the conservation of energy, expressed in the language of quantum mechanics. We now need an expression for the operator corresponding to the other dynamical variable, position. It turns out that the position operator ˆx is simply x itself. 2 This re-interpretation of the dynamical variables as operators leads to a perhaps unexpected result: the product of ˆx and ˆp depends on the order of multiplication. If we define the commutator [ˆr, ŝ] by then by direct calculation [ˆx, ˆp]ψ = x [ˆr, ŝ] = ˆrŝ ŝˆr ( i d ) ( ψ i d ) (xψ) = i x dψ = i ψ + i ψ + i xdψ and we see that [ˆx, ˆp] = i, that is, ˆx and ˆp do not commute. Any good physical theory must be able to predict the result of a given measurement. In quantum mechanics, the measurement of a dynamical variable is predicted by the expectation value of that variable. For example, to determine the expectation value x of position we would write x = + xp (x) where P (x) is the probability density function. Using the Born postulate we rewrite this as x = + ψ (x)xψ(x) Since ˆx = x, we could substitute ˆx for x in the previous equation. Then it is plausible to suppose that the expectation value of momentum is given by p = + ψ (x)ˆpψ(x) 2 Goswami [9, p. 125] puts it cleverly, In its own space, operating by ˆx is the same thing as multiplying by x; at home x doesn t wear a hat.

8 8 JEFF MARSH The result of a measurement must be a real number. Since i appears in the expression for ˆp, how do we know p is real? By direct calculation + ( p p = ψ i dψ ) + ) ψ (i dψ = i + = i ψ ψ + d (ψ ψ) Since ψ(x) is square-integrable we must have ψ (x)ψ(x) 0 as x ±, so the expression vanishes and we conclude that p is real. The operator version of the Schrödinger equation may be further simplified to Ĥψ(x) = Eψ(x) where Ĥ = ˆp2 /2m + V (x) is called the Hamiltonian operator. In this form we see that the Schrödinger equation is an eigenvalue equation. If a given ψ(x) and E satisfy this equation we say that ψ(x) is an eigenfunction (or sometimes eigenstate) of Ĥ with eigenvalue E. 6. The Dirac Notation In the previous section we saw that, in some sense, we can derive the Schrödinger equation by replacing the dynamical variables that appear in the classical equations by differential operators acting on a wave function ψ(x). As was mentioned in section 1, this was the procedure used by Heisenberg in developing his matrix mechanics, only he replaced the variables by infinite-dimensional matrices. Establishing that these two approaches were mathematically equivalent required a new level of abstraction, and to better accommodate this abstraction, Paul A. M. Dirac introduced notations [6] that have become standard tools of the quantum mechanic Welcome to the toolbox. A state of a quantum mechanical system is identified with elements of a complex separable Hilbert space H and denoted using the special symbol. These elements are called kets. The pointy end of the symbol reminds us that elements of a Hilbert space are vectors, and we sometimes call them ket vectors. To specify a particular state by a label, say ψ, we insert it in the middle and write ψ. One of the nice features of this notation is that any label which identifies the quantum state may be written inside the ket symbol. If the state can be identified by an integer i, we may write i. It is not uncommon to see the states of multiple-electron spin systems, where the individual electrons can be either spin-up ( ) or spin-down ( ), written as, for example,. Of course, a linear combination of kets is also a ket, and we may write c 1 ψ 1 + c 2 ψ 2 as c 1 ψ 1 + c 2 ψ 2.

9 TOOLS FOR THE QUANTUM MECHANIC 9 Linear functionals on H are denoted using the bra notation, and a particular linear functional is labeled, for example, φ. As before, the pointy end of the symbol reminds us that linear functionals are vectors. The bra symbol is the mirror-image of the ket symbol, and this reminds us that linear functionals are elements of the dual space H. A bra acting on a ket may be written φ ( ψ ), but the parentheses are redundant and we may write this as φ ψ. This may be further simplified to φ ψ. We say that a bra acting on a ket makes a bra-ket, that is, a bracket. The bracket reminds us of the Mathematician s notation for an inner product, φ, ψ, and indeed, here we use φ ψ to denote the inner product. This notation is justified because the Riesz representation theorem [12, p. 188] tells us that every linear functional can be represented in terms of an inner product. But there is one important difference between these notations. While the Mathematician s inner product φ, ψ is linear in the first argument, and conjugate-linear in the second: c 1 φ 1 + c 2 φ 2, ψ = c 1 φ 1, ψ + c 2 φ 2, ψ φ, c 1 ψ 1 + c 2 ψ 2 = c 1 φ, ψ 1 + c 2 φ, ψ 2 our notation for the inner product φ ψ requires it to be conjugate-linear in the first argument, and linear in the second: c 1 φ 1 + c 2 φ 2 ψ = c 1 φ 1 ψ + c 2 φ 2 ψ φ c 1 ψ 1 + c 2 ψ 2 = c 1 φ ψ 1 + c 2 φ ψ 2 The space H is separable, so there exists a countable orthonormal basis, that is, there is a set of elements { 1, 2, 3,...} such that and any ψ in H can be written as i j = δ ij ψ = i ψ i i The complex coefficients ψ i can be regarded as the coordinates of ψ in the representation defined by { 1, 2, 3,...} and are given by ψ i = i ψ A linear operator  acting on the ket ψ is written as  ψ and produces another ket. Since  is linear we have  c 1 ψ 1 + c 2 ψ 2 = c 1  ψ 1 + c 2  ψ The quantum harmonic oscillator redux. Using our new tools we now present an elegant solution to the quantum harmonic oscillator. First,

10 10 JEFF MARSH with malice aforethought, we introduce the two operators a and a + defined by a mωo = 2 ˆx + iˆp 2m ωo a + = mωo 2 ˆx iˆp 2m ωo It is not hard to show these operators obey the commutation relation Now we calculate ω o a + a = ω o ( mωo 2 ˆx [a, a + ] = 1 = ˆp2 2m mω2 o ˆx iω o[ˆx, ˆp] = Ĥ 1 2 ω o iˆp 2m ωo ) ( mωo 2 ˆx + This result allows us to rewrite the Hamiltonian operator as ( Ĥ = ω o a + a + 1 ) 2 We will also need the following commutation relations Similarly, [Ĥ, a ] = [ ω o a + a, a ] = ω o [a +, a ]a = ω o a [Ĥ, a+ ] = ω o a + ) iˆp 2m ωo Now suppose E is an eigenstate of the Hamiltonian with eigenvalue E Ĥ E = E E Since [Ĥ, a ] = ω o a we may write Substitute for Ĥ E and re-arrange Ĥa E a Ĥ E = ω o a E Ĥa E = (E ω o )a E Thus we conclude that if E is an eigenstate of the Hamiltonian with eigenvalue E, then a E is an eigenstate of the Hamiltonian with eigenvalue E ω o. We call a a lowering operator because it makes a new eigenstate with a lower energy eigenvalue. Similarly, we can show that Ĥa + E = (E + ω o )a + E

11 TOOLS FOR THE QUANTUM MECHANIC 11 and we call a + a raising operator. Collectively, a and a + are called ladder operators. If we have an eigenstate E and apply the lowering operator repeatedly, we create eigenstates of lower energy eigenvalue. However, it turns out that eigenvalues of the harmonic oscillator Hamiltonian must be greater than or equal to zero [9, p. 150]. Therefore, there must be a state E 0 of nonnegative lowest energy, called the ground state. Further application of a cannot lower the energy any more, so we must have a E 0 = 0. Now consider ( Ĥ E 0 = ω o a + a + 1 ) E 0 2 = 1 2 ω o E 0 So we see the energy of the ground state is E 0 = 1 2 ω o, the zero-point energy. Applying the raising operator to E 0 produces a state E 1 with energy E 1 = 3 2 ω o. Can there be a state with energy between E 0 and E 1? No, because applying the lowering operator to this state would make a state with energy lower than the ground state, which is a contradiction. By applying the raising operator n times to E 0 we obtain E n = (n ) ω o, n = 0, 1, 2,... which is the energy quantization condition. References 1. A. B. Arons and M. B. Peppard, Einstein s proposal of the photon concept; a translation of the annalen der physik paper of 1905, American Journal of Physics 33 (1965), no. 5, Max Born, Werner Heisenberg, and Pascual Jordan, Zur quantenmechanik II (On quantum mechanics II), Zeitschrift für Physik 35 (1926), no. 8 9, Max Born and Pascual Jordan, Zur quantenmechanik I (On quantum mechanics I), Zeitschrift für Physik 34 (1925), no. 1, William H. Cropper, The quantum physicists and an introduction to their physics, Oxford University Press, New York, Louis de Broglie, Recherches sur la théorie des quanta (Researches on the quantum theory), Ph.D. thesis, University of Paris, P. A. M. Dirac, The principles of quantum mechanics, fourth ed., Clarendon Press, Oxford, Albert Einstein, Über einen die erzeugung und verwandlung des lichtes betreffenden heuristischen gesichtspunkt (On a heuristic viewpoint concerning the production and transformation of light), Annalen der Physik 322 (1905), no. 6, George Gamow, Thirty years that shook physics; the story of quantum theory, Anchor Books, Garden City, NY, Amit Goswami, Quantum mechanics, second ed., Wm. C. Brown, Dubuque, IA, Werner Heisenberg, Über quantentheoretische umdeutung kinematischer und mechanischer beziehungen (On a quantum-theoretical reinterpretation of kinematic and mechanical relations), Zeitschrift für Physik 33 (1925), no. 1, Thomas F. Jordan, Linear operators for quantum mechanics, Wiley, New York, 1969.

12 12 JEFF MARSH 12. Erwin Kreyszig, Introductory functional analysis with applications, Wiley, New York, Derek F. Lawden, The mathematical principles of quantum mechanics, Methuen, London, Gilbert N. Lewis, The conservation of photons, Nature 118 (1926), no. 2981, Günther Ludwig, Wave mechanics, Pergamon Press, Oxford, George W. Mackey, The mathematical foundations of quantum mechanics; a lecturenote volume, W. A. Benjamin, New York, Max Planck, Über eine verbesserung der wien schen spectralgleichung (On an improvement of Wien s radiation law), Verhandlungen der Deutschen Physikalischen Gesellschaft 2 (1900), no. 13, , Zur theorie des gesetzes der energieverteilung im normalspectrum (On the theory of the energy distribution law of the normal spectrum), Verhandlungen der Deutschen Physikalischen Gesellschaft 2 (1900), no. 17, Eduard Prugovecki, Quantum mechanics in Hilbert space, second ed., Academic Press, New York, Arthur S. Ramsey, Dynamics, Cambridge University Press, Erwin Schrödinger, Quantisierung als eigenwertproblem I (Quantization as an eigenvalue problem I), Annalen der Physik 384 (1926), no. 4, , Quantisierung als eigenwertproblem II (Quantization as an eigenvalue problem II), Annalen der Physik 384 (1926), no. 6, , Quantisierung als eigenwertproblem III (Quantization as an eigenvalue problem III), Annalen der Physik 385 (1926), no. 13, , Quantisierung als eigenwertproblem IV (Quantization as an eigenvalue problem IV), Annalen der Physik 386 (1926), no. 18, John R. Taylor, Scattering theory; the quantum theory of nonrelativistic collisions, Wiley, New York, D. ter Haar, The old quantum theory, Pergamon Press, Oxford, B. L. van der Waerden, Sources of quantum mechanics; edited with a historical introduction, North-Holland Publishing Company, Amsterdam, John von Neumann, Mathematical foundations of quantum mechanics, Princeton University Press, Princeton, NJ, 1955.