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

42	SchroÈdinger wave mechanics
1	1
hpi …ÿ1 pjA( p, t)j2 d p …ÿ1 pA ( p, t)A( p, t) d p		(2:15)
The expectation value	hf ( p)i of any function f ( p) of p is given	by an
expression analogous to equation (2.14)
	1
	hf ( p)i …ÿ1 f ( p)jA( p, t)j2 d p	(2:16)

In general, A( p, t) depends on the time, so that the expectation values hpi and hf ( p)i are also functions of time.

Both Ø(x, t) and A( p, t) contain the same information about the system, making it possible to ®nd hpi using the coordinate-space wave function Ø(x, t) in place of A( p, t). The result of establishing such a procedure will prove useful when determining expectation values for functions of both position and momentum. We begin by taking the complex conjugate of A( p, t) in equation (2.8)

1

1

A ( p, t) p2ð"

…ÿ1Ø (x, t)ei( pxÿEt)=" dx

Substitution of A ( p, t)

into •••••••••

the integral on the right-hand side of equation

(2.15) gives

Ø (x, t) pA( p, t)ei( pxÿEt)=" dx d p

hpi p2ð" ……

1

1

••••••••• ÿ1

1

1

1

…ÿ1Ø (x, t) p2ð" …ÿ1 pA( p, t)ei( pxÿEt)=" d p dx

(2:17)

In order to evaluate the

integral•••••••••

p, we observe that the derivative of

over

Ø(x, t) in equation (2.7), with respect to the position variable x, is

@Ø(x, t)

1

1

i

p2ð" …ÿ1

pA( p, t)ei( pxÿEt)=" d p

Substitution of this

@x

"

•••••••••

observation into equation (2.21) gives the ®nal result

…ÿ1

Ø (x, t)

Ø(x, t) dx

hpi

1

"

@

(2:18)

i

@x

Thus, the expectation value of the momentum can be obtained by an integration in coordinate space.

The expectation value of p2 is given by equation (2.16) with f ( p) p2. The expression analogous to (2.17) is

1	1	1
hp2i …ÿ1Ø (x, t) p2ð"		…ÿ1 p2 A( p, t) ei( pxÿEt)=" d p dx
From equation (2.7) it can be seen•••••••••
	that the quantity in square brackets equals

2.3 Expectation values of dynamical quantities

43

"

2

@

2

Ø(x, t)

i

@x2

so that

…ÿ1

Ø (x, t)

2

hp2i

1

"

@2

Ø(x, t) dx

(2:19)

i

@x2

Similarly, the expectation value of pn is given by

…ÿ1

hpni

1

Ø (x, t)

"

@

nØ(x, t) dx

(2:20)

i

@x

Each of the integrands in equations (2.18), (2.19), and (2.20) is the complex conjugate of the wave function multiplied by an operator acting on the wave function. Thus, in the coordinate-space calculation of the expectation value of the momentum p or the nth power of the momentum, we associate with p the operator ("=i)(@=@x). We generalize this association to apply to the expectation value of any function f ( p) of the momentum, so that

	…ÿ1	Ø (x, t) f "i @@x Ø(x, t) dx
hf ( p)i	1		(2:21)

Equation (2.21) is equivalent to the momentum-space equation (2.16).

We may combine equations (2.14) and (2.21) to ®nd the expectation value of a function f (x, p) of the position and momentum

	…ÿ1	Ø (x, t) f x, "i @@x Ø(x, t) dx
hf (x, p)i	1		(2:22)

Ehrenfest's theorems

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-de®ned wave packet should agree with the classicalmechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest's theorems.

The ®rst relationship is obtained by considering the time dependence of the expectation value of the position coordinate x. The time derivative of hxi in equation (2.13) is

44

dt

SchroÈdinger wave mechanics

j

dt …ÿ1

j

j

…ÿ1

@t j

dhxi

d

1

x

Ø(x,

t) 2 dx

1

x

@

Ø(x, t)

2 dx

1 x

Ø

ÿ Ø

dx

2m

@x

@x

@x

i"

…ÿ1

@

@Ø

@Ø

where equation (2.12) has been used. Integration by parts of the last integral gives

d x

i"

@Ø

@Ø 1

i" 1

@Ø

@Ø

h

i

x Ø

Ø

Ø

Ø

dx

2m

@x ÿ

@x ÿ1ÿ 2m …ÿ1

@x

ÿ

dt

@x

The integrated part vanishes because Ø(x, t) goes to zero as x approaches ( ) in®nity. Another integration by parts of the last term on the right-hand side yields

dhxi		1		1 Ø		" @				Ø dx
	m				i @x
dt				…ÿ1

According to equation (2.18), the integral on the right-hand side of this equation is the expectation value of the momentum, so that we have

h i		dt
p	m	dhxi	(2:23)

Equation (2.23) is the quantum-mechanical analog of the classical de®nition of momentum, p mv m(dx=dt). This derivation also shows that the association in quantum mechanics of the operator ("=i)(@=@x) with the momentum is consistent with the correspondence principle.

The second relationship is obtained from the time derivative of the expecta-

tion value of the momentum hpi in equation (2.18),

dhpi

d

1 Ø

"

@Ø

dx

"

1

@Ø

@Ø

Ø

@ @Ø

dx

…ÿ1

dt

i

dt

…ÿ1

i @x

@t @x

@x @t

We next substitute equations (2.11) for the time derivatives of Ø and Ø and

obtain

dhpi

1

ÿ"2

@2Ø

V Ø

@Ø

Ø

@ "2 @2Ø

V Ø

dx

…ÿ1" 2m @x2

@x

@x 2m @x2

ÿ

dt

#

ÿ"2

1

@2Ø

@Ø

dx

"2

1 Ø

@3Ø

dx

1 Ø Ø

dV

dx

2m

…ÿ1 @x2 @x

2m

…ÿ1

@x3

ÿ …ÿ1

dx

(2:24)

where the terms in V cancel. The ®rst integral on the right-hand side of equation (2.24) may be integrated by parts twice to give

2.3 Expectation values of dynamical quantities

45

1

@2Ø @Ø

Ø @Ø

1

1

@Ø @2Ø

dx

@

ÿ

dx

…ÿ1

@x2

@x

@x

@x

…ÿ1

@x

@x2

@@x

@x

ÿ1

1

Ø @x3

dx

ÿ Ø

@x2

Ø

@Ø

@2Ø

1

@3

Ø

ÿ1

…ÿ1

The integrated part vanishes because Ø and @Ø=@x vanish at ( ) in®nity. The remaining integral cancels the second integral on the right-hand side of equation (2.24), leaving the ®nal result

	dhpi	ÿ	dV	h	F	i	(2:25)
	dt		dx

where equation (2.1) has been used. Equation (2.25) is the quantum analog of Newton's second law of motion, F ma, and is in agreement with the correspondence principle.

Heisenberg uncertainty principle

Using expectation values, we can derive the Heisenberg uncertainty principle introduced in Section 1.5. If we de®ne the uncertainties Äx and Ä p as the standard deviations of x and p, as used in statistics, then we have

Äx h(x ÿ hxi)2i1=2 Ä p h( p ÿ hpi)2i1=2

The expectation values of x and of p at a time t are given by equations (2.13) and (2.18), respectively. For the sake of simplicity in this derivation, we select the origins of the position and momentum coordinates at time t to be the

centers of the wave packet and its

Fourier

transform, so that

hxi 0 and

hpi 0.

The squares of the uncertainties Äx and Ä p are then given by

1

(Äx)2 …ÿ1x2Ø Ø dx

1

" 2

@2Ø

"

2

@Ø

" 2

1

@Ø @Ø

(Ä p)2

1 Ø

dx "

Ø

#

ÿ

dx

i

@x2

i

@x

i

…ÿ1

@x

@x

…ÿ1

ÿ1

"

Ø

" @Ø

1

ÿ

@

dx

i

@x

i

@x

…ÿ1

where the integrated term for (Ä p)2

vanishes because Ø goes to zero as x

approaches ( ) in®nity.2

The product (ÄxÄ p)

is

46 SchroÈdinger wave mechanics

"

Ø

" @Ø

(ÄxÄ p)2

1 (xØ )(xØ) dx

1

ÿ

@

dx

i

@x

i

@x

…ÿ1

…ÿ1

Applying Schwarz's inequality (A.56), we obtain

1

"

1

xØ

@Ø

@Ø

dx

2

"2

1 x

@

(Ø Ø) dx

2

(ÄxÄ p)2 >

xØ

4

i

…ÿ1

@x

@x

4

@x

…ÿ1

2

1

1

2

"

xØ Ø

ÿ …ÿ1Ø Ø dx

4

ÿ1

to zero faster than 1=

x

, as x

The integrated part

vanishes because Ø goes

j j

approaches ( ) in®nity and the remaining integral is unity by

equation (2.9).

p•••••

Taking the square root, we obtain an explicit form of the Heisenberg uncertainty principle

	ÄxÄ p >	"		(2:26)
			2

This expression is consistent with the earlier form, equation (1.44), but relation (2.26) is based on a precise de®nition of the uncertainties, whereas relation (1.44) is not.

2.4 Time-independent SchroÈdinger equation

The ®rst step in the solution of the partial differential equation (2.6) is to

express the wave function Ø(x, t) as the product of two functions
Ø(x, t) ø(x)÷(t)	(2:27)

where ø(x) is a function of only the distance x and ÷(t) is a function of only the time t. Substitution of equation (2.27) into (2.6) and division by the product ø(x)÷(t) give

i"

1 d÷(t)

ÿ

"2 1 d2ø(x)

V(x)

(2:28)

÷(t) dt

2m ø(x) dx2

The left-hand side of equation (2.28) is a function only of t, while the righthand side is a function only of x. Since x and t are independent variables, each side of equation (2.28) must equal a constant. If this were not true, then the left-hand side could be changed by varying t while the right-hand side remained ®xed and so the equality would no longer apply. For reasons that will soon be apparent, we designate this separation constant by E and assume that it is a real number.

Equation (2.28) is now separable into two independent differential equations, one for each of the two independent variables x and t. The time-dependent equation is

2.4 Time-independent SchroÈdinger equation

47

i" d÷d(tt) E÷(t)

which has the solution

÷(t) eÿiEt="

(2:29)

The integration constant in equation (2.29) has arbitrarily been set equal to unity. The spatial-dependent equation is

ÿ	"2 d2ø(x)	V(x)ø(x) Eø(x)	(2:30)
	2m dx2

and is called the time-independent SchroÈdinger equation. The solution of this differential equation depends on the speci®cation of the potential energy V (x). Note that the separation of equation (2.6) into spatial and temporal parts is contingent on the potential V(x) being time-independent.

The wave function Ø(x, t) is then
Ø(x, t) ø(x)eÿiEt="	(2:31)

and the probability density jØ(x, t)j2 is now given by

jØ(x, t)j2 Ø (x, t)Ø(x, t) ø (x)eiEt="ø(x)eÿiEt=" jø(x)j2

Thus, the probability density depends only on the position variable x and does not change with time. For this reason the wave function Ø(x, t) in equation (2.31) is called a stationary state. If Ø(x, t) is normalized, then ø(x) is also normalized

1
…ÿ1jø(x)j2 dx 1	(2:32)

which is the reason why we set the integration constant in equation (2.29) equal to unity.

The total energy, when expressed in terms of position and momentum, is called the Hamiltonian, H, and is given by

H(x, p) p2 V (x) 2m

The expectation value hHi of the Hamiltonian may be obtained by applying equation (2.22)

hHi	…ÿ1
	1	Ø (x, t) ÿ 2"m @@x2 V (x) Ø(x, t) dx
			2	2
For the stationary state (2.31), this expression becomes
hHi		1 ø (x) ÿ	"		@	V (x) ø(x) dx
			2m		@x2
	…ÿ1		2	2

If we substitute equation (2.30) into the integrand, we obtain

48 SchroÈdinger wave mechanics hHi E…1 ø (x)ø(x) dx E

ÿ1

where we have also applied equation (2.32). We have just shown that the separation constant E is the expectation value of the Hamiltonian, or the total energy for the stationary state, so that `E' is a desirable designation. Since the energy is a real physical quantity, the assumption that E is real is justi®ed.

In the application of SchroÈdinger's equation (2.30) to speci®c physical examples, the requirements that ø(x) be continuous, single-valued, and squareintegrable restrict the acceptable solutions to an in®nite set of speci®c functions øn(x), n 1, 2, 3, . . . , each with a corresponding energy value En. Thus, the energy is quantized, being restricted to certain values. This feature is illustrated in Section 2.5 with the example of a particle in a one-dimensional box.

Since the partial differential equation (2.6) is linear, any linear superposition of solutions is also a solution. Therefore, the most general solution of equation

(2.6) for a time-independent potential energy V (x) is
Ø(x, t) Xn	cnøn(x)eÿiEn t="	(2:33)

where the coef®cients cn are arbitrary complex constants. The wave function Ø(x, t) in equation (2.33) is not a stationary state, but rather a sum of stationary states, each with a different energy En.

2.5 Particle in a one-dimensional box

As an illustration of the application of the time-independent SchroÈdinger equation to a system with a speci®c form for V (x), we consider a particle con®ned to a box with in®nitely high sides. The potential energy for such a

particle is given by
V (x) 0,	0 < x < a
1,	x , 0, x . a

and is illustrated in Figure 2.1.

Outside the potential well, the SchroÈdinger equation (2.30) is given by

"2 d2ø

ÿ 2m dx2 1ø Eø

for which the solution is simply ø(x) 0; the probability is zero for ®nding the particle outside the box where the potential is in®nite. Inside the box, the SchroÈdinger equation is

"2 d2ø

ÿ 2m dx2 Eø

2.5 Particle in a one-dimensional box

49

V(x)

x

0

a

Figure 2.1 The potential energy V (x) for a particle in a one-dimensional box of length a.

	or
		d2ø	4ð2			ø
			ÿ						(2:34)
		dx2		ë2
	where ë is the de Broglie wavelength,
			2ð"				h
	We have implicitly assumed here that••••••••••
					p				(2:35)
	ë p2mE

E is not negative. If E were negative, then the wave function ø and its second derivative would have the same sign. As jxj increases, the wave function ø(x) and its curvature d2ø=dx2 would become larger and larger in magnitude and ø(x) would approach ( ) in®nity as x ! 1.

The solutions to equation (2.34) are functions that are proportional to their second derivatives, namely sin(2ðx=ë) and cos(2ðx=ë). The functions exp[2ðix=ë] and exp[ÿ2ðix=ë], which as equation (A.31) shows are equivalent to the trigonometric functions, are also solutions, but are more dif®cult to use

for this system. Thus, the general solution to equation (2.34) is
ø(x) A sin	2ðx	B cos	2ðx		(2:36)
	ë		ë

where A and B are arbitrary constants of integration.

The constants A and B are determined by the boundary conditions placed on the solution ø(x). Since ø(x) must be continuous, the boundary conditions require that ø(x) vanish at each end of the box so as to match the value of ø(x) outside the box, i.e., ø(0) ø(a) 0. At x 0, the function ø(0) from (2.36)

	is
	ø(0) A sin 0 B cos 0 B
	so that B 0 and ø(x) is now	2ðx
	ø(x) A sin		(2:37)
		ë

At x a, ø(a) is

50	SchroÈdinger wave mechanics
	ø(a) A sin	2ða	0
		ë

The constant A cannot be zero, for then ø(x) would vanish everywhere and there would be no particle. Consequently, we have sin(2ða=ë) 0 or

2ða	nð,	n 1, 2, 3, . . .
ë

where n is any positive integer greater than zero. The solution n 0 would cause ø(x) to vanish everywhere and is therefore not acceptable. Negative values of n give redundant solutions because sin(ÿè) equals ÿsin è.

We have found that only distinct values for the de Broglie wavelength satisfy the requirement that the wave function represents the motion of the particle.

These distinct values are denoted as ën and are given by
ën	2a	,	n 1, 2, 3, . . .	(2:38)
	n

Consequently, from equation (2.35) only distinct values En of the energy are allowed

En	n2ð2"2		n2 h2		n 1, 2, 3, . . .	(2:39)
				,
	2ma2		8ma2

so that the energy for a particle in a box is quantized.

The lowest allowed energy level is called the zero-point energy and is given by E1 h2=8ma2. This zero-point energy is always greater than the zero value of the constant potential energy of the system and increases as the length a of the box decreases. The non-zero value for the lowest energy level is related to the Heisenberg uncertainty principle. For the particle in a box, the uncertainty Äx in position is equal to the length a since the particle is somewhere within the box. The uncertainty Ä p in momentum is equal to 2jpj since the momentum ranges from ÿjpj to jpj. The momentum and energy are related by

so that	j		p	••••••••••		nh
			j			2a
		p		2mE

ÄxÄ p nh

is in agreement with the Heisenberg uncertainty principle (2.26). If the lowest allowed energy level were zero, then the Heisenberg uncertainty principle would be violated.

The allowed wave functions øn(x) for the particle in a box are obtained by

substituting equation (2.38) into (2.37),
øn(x) A sin	nðx	,	0 < x < a
	a

(2:42)

2.5 Particle in a one-dimensional box

51

The remaining constant of integration A is determined by the normalization condition (2.32),

1

a

nðx

a

ð

a

…ÿ1jøn(x)j2 dx jAj2

…0 sin2

dx jAj2

…0 sin2nè dè jAj2

1

a

ð

2

where equation (A.15) was used. Therefore, we have

2

jAj2

a

or

A

eiá

2

a

r•••

Setting the phase á equal to zero since it has no physical signi®cance, we

obtain for the normalized wave functions
øn(x)		2	sin	nðx	,	0 < x < a
							(2:40)
		a		a
	r•••					x , 0, x . a
	0,

The time-dependent SchroÈdinger equation (2.30) for the particle in a box has an in®nite set of solutions øn(x) given by equation (2.40). The ®rst four wave functions øn(x) for n 1, 2, 3, and 4 and their corresponding probability densities jøn(x)j2 are shown in Figure 2.2. The wave function ø1(x) corresponding to the lowest energy level E1 is called the ground state. The other wave functions are called excited states.

If we integrate the product of two different wave functions øl(x) and øn(x),

we ®nd that

sin

a

2

a

lðx

nðx

dx

2 ð

sin lè sin nè dè 0

…

0

øl(x)øn(x) dx a

…

0

sin a

a

ð

…

0

(2:41)

where equation (A.15) has been introduced. This result may be combined with

the normalization relation to give

…a

øl(x)øn(x) dx äln

0

where äln is the Kronecker delta,

äln 1,	l n
0,	(2:43)
	l 6n

Functions that obey equation (2.41) are called orthogonal functions. If the orthogonal functions are also normalized, as in equation (2.42), then they are