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1 VI. OPERATORS & EXPECTATION VALUES 1. The Calculation of Average Values The description of quantum states in terms of probability amplitudes at once implies the possibility of using these probabi lity amplitudes to calculate the average values of various quanti ties for systems that are in a a specified state. Consider, for example, a bound state of a particle in a one-dimensional poten tial. Such a state has a unique value of the energy,'but there is a continuous distribution of the possible values of the position, x. The probability amplitude associated with a particular value of x is given by 4>(x). The probability that the particle will be! I 2 found between x and x + dx is then equal to j^(x)[ dx, provided that tjkx) is normalized in the usual way: X - I (6.1) If we ask "What is the average position of a particle described by ikx)?"- w^ can supply the answer by evaluating a weighted mean value, just as in classical probability calculations. We multiply each particular value of x by the probability of having that value, and sum (integrate) over all possibilities: *, Such averages are called expectation values in quantum mechanics. This name vividly expresses the essentially statistical nature o:5 ; -1-

2 -2- the quantum-mechanical description. One can imagine a huge number of identical systems, each consisting of a particle in the state described by ip(x). If one carried out a set of experiments to determine the location of each of these particles, the results would be distributed in accordance with the relative 2 The result of an individual observation is not predictable, but the mean of all the measurements is well defined. It is customary to denote the expectation value of any quantity Q by enclosing it within angular brackets ; thus we put Expectation value of x (6.2) 1 To take a specific example, consider any one bound state of a particle in a "violin-string" state: VI The expectation value of x for this state is then given by L? C ' 2- t J X Sm O It is easy to verify that the result of this calculation is,that (x^ is equal to L/2 for any n. Notice that the expectation value of a quantity is in principle not at all the same thing as the most probable value. For example, if we are concerned with measurements of position, the most probable value of x is the value / Iffi

3 -3- at which J«y(x) ( is greatesti If we consider the violin-string states u/(x), then for n = 1 the expectation value and the most probable value do coincide, but for n = 2 the value of is zero; thus a particle in this state would never be found at a value of x precisely equal to C^. *N Given that I MX) j dx is the probability associated with a particular value of x, we can use it to calculate the expecta tion value of any function of x.. We simply have <fm> - Thus we can, for example, find the mean squared value of x 2 2. Expectation Values involving Operators Suppose that we again consider any individual energy-state of a particle in a one-dimensional box, and this time ask what is the expectation value of the linear momentum p. A consideration j^ of the symmetry of the situation suggests, without the need for any calculation, that the answer is zero; the particle is equally likely to be found moving to right or left. We can spell this out more formally by noting that the wave function can be written as a superposition, with equal weights, of plane-wave amplitudes belonging to equal and opposite values of the wave-number k : m km x - fit - ne I yv rt /\ Thus k = + k, and the momentum p (=ftk) is equal to + ^k n *x ^ n This makes it clear that the probability associated with each

4 4 value of the two values of p is 1/2, and so we have <px> = [Notice, again, the distinction between expectation value and most probable value. Individual measurements on this state would never yield a value of k equal to zero, but only + k.] The above calculation is very simple and straightforward, but this is mainly because the situation itself is very simple. What we shall now do is to introduce a method of using the Schrfidinger amplitude itself to calculate ^p > and related quantities. With its help we can then handle problems that are far less obvious than the one we are now using as an example. The starting point is the following very important concept: In quantum mechanics we can associate various dynamical quantities with particular mathematical operators. This is a rather general statement, but we shall at once make it explicit in connection with linear momentum. We have seen that the probability amplitude for a free particle of given momentum v is of the form /1 t(k*~ - ft e where k = p /fi. Thus we can put or We then propose that, in general, the operator *~t*\c0fbty r operating

5 -5- on a Schrodinger amplitude ip, is equivalent to multiplying by p. Thus we put fi (6-3) We then proceed to evaluate the expectation value of p by means of the following equation: < (6.4) If, as in any one of the bound states of a particle in a box, the SchrOdinger amplitude $ embodies contributions from more than one value of ' p, the mathematical operator corresponding to p, *k «V applied to J, yields all these component momenta and the integral in eq. (6.4) is an automatic prescription for forming the appropriate weighted average. It is important to observe the order of the various factors in this integral. Now that p is represented j^> by a differential operator/ applying to anything that follows it, we must place the factor $* to its left so as to make clear that it is not subject to this operation. Suppose once again that ip represents a particular energy- state of a particle in a one-dimensional box: Then 2. L o

6 -6- But the functions sin k x and cos k x are orthogonal (i.e., j sin k x cos k x dx = 0), so this calculation gives vp / = 0 0 J n n ^ N ^x as we know it must. However, a calculation of p by the same technique gives a non-zero result. The operator correspending to p jf± is -# ( d /^x ), and so for this same state we have ^ Nr 2 w 2 ' f (sin ' knx)[^/sin fv\ krx) dx / \. ty, 2 i.e., <px > = V kn 2 f L 2 L T n J f sin k n x dx 0 This is just what we expect, since the (kinetic) energy E of the state will be given by <P 2 > F n n 2m 2m With k = ntt/l, this gives us the familiar result 2 2 n h n 8mL 2 We could, of course, have obtained this expectation value of p wrv much more readily by going back to the description of the state as a. 50:50 mixture of the two momentum components Ik. On this basis we have simply <PX2 > = K 2 <k2 > = # 2 [ i(kn ) 2 + i(-kn ) 2] = Thus, up to this point, our use of the momentum operator technique has been like the proverbial sledge-hcmmer to kill a fly. But let us now take an example where it is not. so trivial.

7 -7- Consider a particle in one of the characteristic energy states in a harmonicoscillator potential. Now the value of p, as defined x-0 by + /2m[E - V(x)J, takes on an infinite number of different values (including imaginary ones). Given a knowledge of 1 /, we can however evaluate with the help of eq. (6.4). Suppose that the state in question is the lowest state of the oscillator The spatial factor in its wave function is of the form = A e 2/0 2 -x /2a (6.5a) where 2 a =s mc [cf. Notes, Ch. Ill, pp. 19 ff.] The normalization of \ / leads to the condition The value- of A2 = a ATT ) is given by (6.5b) <PX> - -in K A *2 J C e -* / /2a = %- j» "CO 2~, 2 - x /a dx d_ dx Since the integrand is the product of an odd function (x itself) with an even function, the integral between the limits + oo i s zero. This is another result that could have been expected.

8 -8- y 2 v More interesting, perhaps, is the calculation of \p >. For * this we have <px2 > = -*2A2 f v /Oa inl v /Oa X//:d U,^A/Zcl x ^^ dx ^2A2 f -x 2/a e ' dx - A' 4 (30 o 2 y 2 2 -x /a, x e 7 dx 2a 2 Substituting the value of A from eq.(6.5b), this gives us <PX 2 > = - (6.6) 2a The calculation of expectation values of the energy E can likewise be based on the use of an equivalent operator. Referring again in the first instance to a free-particle wave function, we have and hence TV i (kx-et/fi) = Ae N ' or E <^-> ih ^ (6.7) ^t If, now, we take a wave-function that represents a superposition of different energy-states, we have e-^n*/* ifl Then E = C f /f \/ _ 7 ' ' E

9 -9- If the different ( f s are orthogonal and normalized, this reduces to the equation < E >= ra n Thus/ again as one might expect/ the value of ^E^ is just a weighted average of the energies of the component states, with weighting factors equal to the integrated probabilities associated with the individual components. In some circumstances it may be of interest to calculate the mean values of the kinetic energy and the potential energy separately for a particle in a given state. The ways of evaluat ing these quantities are essentially contained in what we have already done. The operator for the kinetic energy K is the 2 operator corresponding to p /2m ; thus for one-dimensional problems we put K ~~ (6-8a) and for three dimensions we have In a one-dimensional system we therefore have K <r~> -_ (6.8b) For the mean value of the potential energy/ we simply use the procedure that applies to an arbitrary function of position, so that V(x) I//" <^x (6.10)

10 -10 If we consider the (one-dimensional) Schrodinger equation as it applies to an individual energy-state, we have dx -* Multiplying by \^ and integrating over all x then leads, with the help of eqs. (6.9) and (6.10), to the reasonable result <K> + <V> = E (6.11) 3. Variances and Uncertainty Relations In statistical analysis, when one is dealing with a quantity that has a certain distribution of values, an automatic measure of the spread of the distribution around its mean value is provided by the standard deviation, CT. The square of CT, known as the variance, is the mean squared deviation of the individual values from the mean. Thus for some arbitrary quantity Q, we have Variance = -CT 2 = (Q - <Q> ) 2 >C ClV 4 But (Q - <Q> ) 2 = Q 2-2Q <Q> + <Q> 2, and the variance is the expectation value of this expression. This gives <rq2 = <Q2 > - 2 <Q><Q> 4- <Q> 2 * i.e., <TQ2 = <Q2 > - <Q> 2 (6.12) tf Only if Q is limited to a single sharp value does the variance 2 drop to zero; in all other cases the difference between t and Q is positive.

11 -11- We can use this measure of the width of a probability distribution to examine the uncertainty relations in more specific terms. Consider, for example, the lowest state of the harmonic oscillator, as described by eqs.(6.5). For this we have = o fir a 3 ^ 2, Thus CT 2 Now in Section 2 we calculated the values of p > and J\. the results were <PX> =o, 2, -h 2 <PV > = 2 a' 2 Hence h 2 2 a 2 It follows, then, that the following relation holds: x Px = ^ (6.13) This uncertainty product is thus independent of the parameter a that characteri2es the width of the position probability distribu tion. It happers that the value fc/2, characteristic of these

12 -12- Gaussian error-function probability distributions, represents the smallest achievable value of Ap «/\ x, where A p and Ax are defined as the standard deviations of the probability distributions in question. [4. The Quantum-Mechanical Equivalent of Newton's Law An interesting application of the calculation of expecta tion values is the formulation of an equation that parallels the basic law of motion in classical dynamics. We begin with the expectation value of p as defined in eq. (6.4), and then con- «sider its time derivative: Now with the help of the time-dependent Schrfldinger equation we can convert the integrands on the right into forms that do not explicitly involve t:. w it *

13 -13- Hence at -JL 2.w fi J ^ For any reasonable wave- function, the values of \ and its derivatives vanish at x = + cx> Thus we are left with the result Since the negative gradient of the potential energy is equal to the force derived from that potential, eg. (6.14) corresponds to a statement of F = dp/dt in terms of the expectation values of these quantities. If one wants to carry the analysis a stage further back, one can show by similar methods that the following result also holds: m ft < x> = <' px>

14 Expectation Values in Superpositions of States If we have a superposition of two different energy-states of a quantum-mechanical system, there is a harmonic time-dependence of the expectation values of position and related quantities. Suppose that the state in question is given by. Then + f 2-f./il/,^x) VMd*} cos cot where ^0 = (E 2 - Ej)/K Each of the three integrals in the above expression is a definite integral representing a quantity of the dimension of length. Thus the equation for ^x> can be written in the simplified form <x> = A + B cos 60 1 (6.16) If the wave-function in question describes an electrically charged particle, one can see here the basis of a picture of an oscillating electric dipole, with the implied possibility of associated v radiation characterized by the frequency 0).

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