Fitts D.D. - Principles of Quantum Mechanics[c] As Applied to Chemistry and Chemical Physics (1999)(en)

32

The wave function

eijØB (x)ØA(x)

Pj(x)

j

ØA(x) 2

j

ØB(x)

2

eÿijØA(x)ØB

(x)

j

j

PA(x) PB(x) I j(x)

(1:51)

where I j(x) is de®ned by

I j(x)

eÿijØA(x)ØB(x)

eijØB(x)ØA(x)

The interaction with the detector at slit A has changed the interference term from I AB(x) to I j(x).

For any particular particle leaving the source S and ultimately striking the detection screen D, the value of j is determined by the interaction with the detector at slit A. However, this value is not known and cannot be controlled; for all practical purposes it is a randomly determined and unveri®able number. The value of j does, however, in¯uence the point x where the particle strikes the detection screen. The pattern observed on the screen is the result of a large number of impacts of particles, each with wave function Ø(x) in equation (1.50), but with random values for j. In establishing this pattern, the term I j(x) in equation (1.51) averages to zero. Thus, in this experiment the probability density Pj(x) is just the sum of PA(x) and PB(x), giving the intensity distribution shown in Figure 1.9(b).

In comparing the two experiments with both slits open, we see that interacting with the system by placing a detector at slit A changes the wave function of the system and the experimental outcome. This feature is an essential characteristic of quantum theory. We also note that without a detector at slit A, there are two indistinguishable ways for the particle to reach the detection screen D and the two wave functions ØA(x) and ØB(x) are added together. With a detector at slit A, the two paths are distinguishable and it is the probability densities PA(x) and PB(x) that are added.

An analysis of the Stern±Gerlach experiment also contributes to the interpretation of the wave function. When an atom escapes from the hightemperature oven, its magnetic moment is randomly oriented. Before this atom interacts with the magnetic ®eld, its wave function Ø is the weighted sum of

two possible states á and â
Ø cáá cââ	(1:52)
where cá and câ are constants and are related by
jcáj2 jcâj2 1

In the presence of the inhomogeneous magnetic ®eld, the wave function Ø collapses to either á or â with probabilities jcáj2 and jcâj2, respectively. The state á corresponds to the atomic magnetic moment being parallel to the magnetic ®eld gradient, the state â being antiparallel. Regardless of the

1.8 Physical interpretation of the wave function

33

orientation of the magnetic ®eld gradient, vertical (up or down), horizontal (left or right), or any angle in between, the wave function of the atom is always given by equation (1.52) with á parallel and â antiparallel to the magnetic ®eld gradient. Since the atomic magnetic moments are initially randomly oriented, half of the wave functions collapse to á and half to â.

In the Stern±Gerlach experiment with two magnets having parallel magnetic ®eld gradients±the `®rst arrangement' described in Section 1.7±all the atoms entering the second magnet are in state á and therefore are all de¯ected in the same direction by the second magnetic ®eld gradient. Thus, it is clear that the wave function Ø before any interaction is permanently changed by the interaction with the ®rst magnet.

In the `second arrangement' of the Stern±Gerlach experiment, the atoms emerging from the ®rst magnet and entering the second magnet are all in the same state, say á. (Recall that the other beam of atoms in state â is blocked.) The wave function á may be regarded as the weighted sum of two states á9 and â9

á c9áá9 c9ââ9

where á9 and â9 refer to states with atomic magnetic moments parallel and antiparallel, respectively, to the second magnetic ®eld gradient and where c9á and c9â are constants related by

jc9áj2 jc9âj2 1

In the `second arrangement', the second magnetic ®eld gradient is perpendicular to the ®rst, so that

jc9áj2 jc9âj2 12

and

1

á p•••(á9 â9) 2

The interaction of the atoms in state á with the second magnet collapses the wave function á to either á9 or â9 with equal probabilities.

In the `third arrangement', the right beam of atoms emerging from the second magnet (all atoms being in state á9), passes through a third magnetic ®eld gradient parallel to the ®rst. In this case, the wave function á9 may be expressed as the sum of states á and â

1

á9 p•••(á â) 2

The interaction between the third magnetic ®eld gradient and each atom collapses the wave function á9 to either á or â with equal probabilities.

The interpretation of the various arrangements in the Stern±Gerlach experi-

34	The wave function

ment reinforces the postulate that the wave function for a particle is the sum of indistinguishable paths and is modi®ed when the paths become distinguishable by means of a measurement. The nature of the modi®cation is the collapse of the wave function to one of its components in the sum. Moreover, this new collapsed wave function may be expressed as the sum of subsequent indistinguishable paths, but remains unchanged if no further interactions with measuring devices occur.

This statistical interpretation of the signi®cance of the wave function was postulated by M. Born (1926), although his ideas were based on some experiments other than the double-slit and Stern±Gerlach experiments. The concepts that the wave function contains all the information known about the system it represents and that it collapses to a different state in an experimental observation were originated by W. Heisenberg (1927). These postulates regarding the meaning of the wave function are part of what has become known as the Copenhagen interpretation of quantum mechanics. While the Copenhagen interpretation is disputed by some scientists and philosophers, it is accepted by the majority of scientists and it provides a consistent theory which agrees with all experimental observations to date. We adopt the Copenhagen interpretation of quantum mechanics in this book.3

Problems

1.1 The law of dispersion for surface waves on a sheet of water of uniform depth d is4

ù(k) ( gk tanh dk)1=2

where g is the acceleration due to gravity. What is the group velocity of the resultant composite wave? What is the limit for deep water (dk > 4)?

1.2The phase velocity for a particular wave is vph A=ë, where A is a constant. What is the dispersion relation? What is the group velocity?

1.3Show that

…1

A(k) dk 1

ÿ1

for the gaussian function A(k) in equation (1.19).

3The historical and philosophical aspects of the Copenhagen interpretation are more extensively discussed in J. Baggott (1992) The Meaning of Quantum Theory (Oxford University Press, Oxford).

4For a derivation, see H. Lamb (1932) Hydrodynamics, pp. 363±81 (Cambridge University Press, Cambridge).

Problems

35

1.4Show that the average value of k is k0 for the gaussian function A(k) in equation (1.19).

1.5Show that the gaussian functions A(k) and Ø(x, t) obey Parseval's theorem (1.18).

1.6Show that the square pulse A(k) in equation (1.21) and the corresponding function Ø(x, t) obey Parseval's theorem.

2

SchroÈdinger wave mechanics

2.1 The SchroÈdinger equation

In the previous chapter we introduced the wave function to represent the motion of a particle moving in the absence of an external force. In this chapter we extend the concept of a wave function to make it apply to a particle acted upon by a non-vanishing force, i.e., a particle moving under the in¯uence of a potential which depends on position. The force F acting on the particle is

related to the potential or potential energy V(x) by
dV
F ÿ			(2:1)
	dx

As in Chapter 1, we initially consider only motion in the x-direction. In Section 2.7, however, we extend the formalism to include three-dimensional motion.

In Chapter 1 we associated the wave packet
	1	1
Ø(x,	t) p2ð" …ÿ1 A( p)ei( pxÿEt)=" d p		(2:2)
with the motion in the x-	•••••••••
	direction of a free particle, where the weighting factor
A( p) is given by
	1	1
A( p) p2ð"		…ÿ1Ø(x, t)eÿi( pxÿEt)=" dx	(2:3)
This wave packet satis®es a •••••••••
	partial differential equation, which will be used as

the basis for the further development of a quantum theory. To ®nd this differential equation, we ®rst differentiate equation (2.2) twice with respect to the distance variable x to obtain

	@x2 p2ð"5			…ÿ1
@2Ø			1	1
			ÿ	p2 A( p)ei( pxÿEt)=" d p	(2:4)
Differentiation of (2.2) with			••••••••••
			respect to the time t gives

36

2.2

The wave function

37

@t p2ð"3 …ÿ1

@Ø

ÿi

1 EA( p)ei( pxÿEt)=" d p

(2:5)

The total energy E for a free

••••••••••

particle (i.e., for a particle moving in a region of

constant potential energy V ) is given by

p2

E

V

2m

which may be combined with equations (2.4) and (2.5) to give

i"

@Ø

"2 @2Ø

V Ø

ÿ

@t

2m

@x2

SchroÈdinger (1926) postulated that this differential equation is also valid when the potential energy is not constant, but is a function of position. In that

case the partial differential equation becomes
i"	@Ø(x, t)		ÿ	"2 @2Ø(x, t)				V(x)Ø(x, t)	(2:6)
	@t			2m @x2

which is known as the time-dependent SchroÈdinger equation. The solutions

Ø(x, t) of equation (2.6) are the time-dependent wave functions. An important goal in wave mechanics is solving equation (2.6) for Ø(x, t) using various expressions for V (x) that relate to speci®c physical systems.

When V(x) is not constant, the solutions Ø(x, t) to equation (2.6) may still

be expanded in the form of a wave packet,
1	1
Ø(x, t) p2ð"	…ÿ1 A( p, t)ei( pxÿEt)=" d p	(2:7)
The Fourier transform A( p, t) •••••••••
is then, in general, a function of both p and time
t, and is given by
1	1
A( p, t) p2ð" …ÿ1Ø(x, t)eÿi( pxÿEt)=" dx		(2:8)

•••••••••

By way of contrast, recall that in treating the free particle as a wave packet in Chapter 1, we required that the weighting factor A( p) be independent of time and we needed to specify a functional form for A( p) in order to study some of the properties of the wave packet.

2.2 The wave function

Interpretation

Before discussing the methods for solving the SchroÈdinger equation for speci®c choices of V(x), we consider the meaning of the wave function. Since the wave function Ø(x, t) is identi®ed with a particle, we need to establish the connection between Ø(x, t) and the observable properties of the particle. As in the

38	SchroÈdinger wave mechanics

case of the free particle discussed in Chapter 1, we follow the formulation of Born (1926).

The fundamental postulate relating the wave function Ø(x, t) to the properties of the associated particle is that the quantity jØ(x, t)j2 Ø (x, t)Ø(x, t) gives the probability density for ®nding the particle at point x at time t. Thus, the probability of ®nding the particle between x and x dx at time t is jØ(x, t)j2 dx. The location of a particle, at least within an arbitrarily small interval, can be determined through a physical measurement. If a series of measurements are made on a number of particles, each of which has the exact same wave function, then these particles will be found in many different locations. Thus, the wave function does not indicate the actual location at which the particle will be found, but rather provides the probability for ®nding the particle in any given interval. More generally, quantum theory provides the probabilities for the various possible results of an observation rather than a precise prediction of the result. This feature of quantum theory is in sharp contrast to the predictive character of classical mechanics.

According to Born's statistical interpretation, the wave function completely describes the physical system it represents. There is no information about the system that is not contained in Ø(x, t). Thus, the state of the system is determined by its wave function. For this reason the wave function is also called the state function and is sometimes referred to as the state Ø(x, t).

The product of a function and its complex conjugate is always real and is positive everywhere. Accordingly, the wave function itself may be a real or a complex function. At any point x or at any time t, the wave function may be positive or negative. In order that jØ(x, t)j2 represents a unique probability density for every point in space and at all times, the wave function must be continuous, single-valued, and ®nite. Since Ø(x, t) satis®es a differential equation that is second-order in x, its ®rst derivative is also continuous. The wave function may be multiplied by a phase factor eiá, where á is real, without changing its physical signi®cance since

[eiáØ(x, t)] [eiáØ(x, t)] Ø (x, t)Ø(x, t) jØ(x, t)j2

Normalization

The particle that is represented by the wave function must be found with probability equal to unity somewhere in the range ÿ1 < x < 1, so that

Ø(x, t) must obey the relation

…1

jØ(x, t)j2 dx 1

(2:9)

ÿ1

2.2 The wave function

39

A function that obeys this equation is said to be normalized. If a function Ö(x, t) is not normalized,…but1 satis®es the relation

Ö (x, t)Ö(x, t) dx N

ÿ1

then the function Ø(x, t) de®ned by

Ø(x, t) p1••••• Ö(x, t)

N

is normalized.

In order for Ø(x, t) to satisfy equation (2.9), the wave function must be

square-integrable (also called quadratically integrable). Therefore, Ø(x, t) p••••••

must go to zero faster than 1= jxj as x approaches ( ) in®nity. Likewise, the derivative @Ø=@x must also go to zero as x approaches ( ) in®nity.

Once a wave function Ø(x, t) has been normalized, it remains normalized as

time progresses. To prove this assertion, we consider the integral
N	…ÿ1Ø Ø dx
	1

and show that N is independent of time for every function Ø that obeys the SchroÈdinger equation (2.6). The time derivative of N is

dN	1 @
	…ÿ1		jØ(x, t)j2 dx	(2:10)
dt		@t

where the order of differentiation and integration has been interchanged on the right-hand side. The derivative of the probability density may be expanded as follows

@

2

@

@Ø

@Ø

jØ(x, t)j

(Ø Ø) Ø

Ø

@t

@t

@t

@t

Equation (2.6) and its complex conjugate may be written in the form

@Ø

i" @2Ø

i

ÿ

V Ø

@t

2m

@x2

"

(2:11)

@Ø

i" @2Ø

i

ÿ

V Ø

@t

2m

@x2

"

so that @jØ(x, t)j2=@t becomes

ÿ Ø

@x2

@t jØ(x, t)j2

2m Ø @x2

@

i"

@2Ø

@2Ø

where the terms containing V cancel. We next note that

@x2

@x Ø

@x

ÿ Ø @x

Ø

@x2 ÿ Ø

@

@Ø

@Ø

@2Ø

@2Ø

40

SchroÈdinger wave mechanics

so that

Ø

@

i" @

@Ø

@Ø

jØ(x, t)j2

ÿ Ø

(2:12)

@t

2m @x

@x

@x

Substitution of equation (2.12) into (2.10) and evaluation of the integral give

dN i"

1

@

@Ø @Ø

i"

@Ø @Ø

1

Ø

ÿ Ø

dx

Ø

ÿ Ø

dt

2m

…ÿ1

@x

@x

@x

2m

@x

@x

ÿ1

Since Ø(x, t) goes to zero as x goes to ( ) in®nity, the right-most term vanishes and we have

ddNt 0

Thus, the integral N is time-independent and the normalization of Ø(x, t) does not change with time.

Not all wave functions can be normalized. In such cases the quantity

jØ(x, t)j2 may be regarded as the relative probability density, so that the ratio

…a2

jØ(x, t)j2 dx

…a1

b2

jØ(x, t)j2 dx

b1

represents the probability that the particle will be found between a1 and a2 relative to the probability that it will be found between b1 and b2. As an example, the plane wave

Ø(x, t) ei( pxÿEt)="

does not approach zero as x approaches ( ) in®nity and consequently cannot be normalized. The probability density jØ(x, t)j2 is unity everywhere, so that the particle is equally likely to be found in any region of a speci®ed width.

Momentum-space wave function

The wave function Ø(x, t) may be represented as a Fourier integral, as shown in equation (2.7), with its Fourier transform A( p, t) given by equation (2.8). The transform A( p, t) is uniquely determined by Ø(x, t) and the wave function Ø(x, t) is uniquely determined by A( p, t). Thus, knowledge of one of these functions is equivalent to knowledge of the other. Since the wave function Ø(x, t) completely describes the physical system that it represents, its Fourier transform A( p, t) also possesses that property. Either function may serve as a complete description of the state of the system. As a consequence, we may interpret the quantity jA( p, t)j2 as the probability density for the momentum at

2.3 Expectation values of dynamical quantities

41

time t. By Parseval's theorem (equation (B.28)), if Ø(x, t) is normalized, then its Fourier transform A( p, t) is normalized,

1	1
…ÿ1jØ(x, t)j2 dx	…ÿ1jA( p, t)j2 d p 1

The transform A( p, t) is called the momentum-space wave function, while Ø(x, t) is more accurately known as the coordinate-space wave function. When there is no confusion, however, Ø(x, t) is usually simply referred to as the wave function.

2.3 Expectation values of dynamical quantities

Suppose we wish to measure the position of a particle whose wave function is Ø(x, t). The Born interpretation of jØ(x, t)j2 as the probability density for ®nding the associated particle at position x at time t implies that such a measurement will not yield a unique result. If we have a large number of particles, each of which is in state Ø(x, t) and we measure the position of each of these particles in separate experiments all at some time t, then we will obtain a multitude of different results. We may then calculate the average or mean value hxi of these measurements. In quantum mechanics, average values of dynamical quantities are called expectation values. This name is somewhat misleading, because in an experimental measurement one does not expect to obtain the expectation value.

By de®nition, the average or expectation value of x is just the sum over all possible values of x of the product of x and the probability of obtaining that value. Since x is a continuous variable, we replace the probability by the

probability density and the sum by an integral to obtain

…1

hxi	xjØ(x, t)j2 dx	(2:13)
	ÿ1
More generally, the expectation value hf (x)i of any function		f (x) of the
variable x is given by
	1
hf (x)i …ÿ1 f (x)jØ(x, t)j2 dx		(2:14)

Since Ø(x, t) depends on the time t, the expectation values hxi and hf (x)i in equations (2.13) and (2.14) are functions of t.

The expectation value hpi of the momentum p may be obtained using the momentum-space wave function A( p, t) in the same way that hxi was obtained from Ø(x, t). The appropriate expression is