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

52

ø1

0

ø2

0

ø3

0

ø4

0

SchroÈdinger wave mechanics

|ø1|2

x

a

0

a

|ø2|2

x

a

0

a

|ø3|2

x

a

0

a

|ø4|2

x

a

0

a

Figure 2.2 Wave functions øi and probability densities jøij2 for a particle in a onedimensional box of length a.

said to be orthonormal. The orthogonal property of wave functions in quantum mechanics is discussed in a more general context in Section 3.3.

The stationary states Ø(x, t) for the particle in a one-dimensional box are

given by substitution of equations (2.39) and (2.40) into (2.31),
Ø(x, t)		2	sin		nðx		eÿi(n2ð2"=2ma2)t	(2:44)
					a
		a
	r•••

The most general solution (2.33) is, then,

Ø(x, t)	2				cn sin
		a		n
	r•••
				X

nðx	2	2	2
a	eÿi(n	ð	"=2 ma	)t	(2:45)

2.6 Tunneling	53
2.6 Tunneling

As a second example of the application of the SchroÈdinger equation, we consider the behavior of a particle in the presence of a potential barrier. The speci®c form that we choose for the potential energy V (x) is given by

V (x) V0,	0 < x < a
0,	x , 0, x . a
and is shown in Figure 2.3. The	region where x , 0 is labeled I, where

0 < x < a is labeled II, and where x . a is labeled III.

Suppose a particle of mass m and energy E coming from the left approaches the potential barrier. According to classical mechanics, if E is less than the barrier height V0, the particle will be re¯ected by the barrier; it cannot pass through the barrier and appear in region III. In quantum theory, as we shall see, the particle can penetrate the barrier and appear on the other side. This effect is called tunneling.

In regions I and III, where V (x) is zero, the SchroÈdinger equation (2.30) is

d2ø(x)			2mE
			ÿ		ø(x)	(2:46)
	dx2			"2
The general solutions to equation (2.46) for these regions are
	øI Aeiáx Beÿiáx					(2:47 a)
øIII Feiáx Geÿiáx						(2:47 b)

where A, B, F, and G are arbitrary constants of integration and á is the

abbreviation		p
		••••••••••
á		2mE	(2:48)
		"

In region II, where V(x) V0 . E, the SchroÈdinger equation (2.30) becomes

V(x)
V0
I			II		III
						x
0
				a

Figure 2.3 Potential energy barrier of height V0 and width a.

54		SchroÈdinger wave mechanics
	d2ø(x)					2m
							(V0 ÿ E)ø(x)	(2:49)
		dx2				"2
for which the general solution is
		øII Ceâx Deÿâx						(2:50)
where C and D are integration constants and â is the abbreviation
	â					2m(V0 ÿ E)		(2:51)
						"
			p••••••••••••••••••••••••
The	term exp[iáx] in equations					(2.47) indicates travel in the		positive x-

direction, while exp[ÿiáx] refers to travel in the opposite direction. The coef®cient A is, then, the amplitude of the incident wave, B is the amplitude of the re¯ected wave, and F is the amplitude of the transmitted wave. In region III, the particle moves in the positive x-direction, so that G is zero. The relative probability of tunneling is given by the transmission coef®cient T

	T	jFj2	(2:52)
		jAj2

and the relative probability of re¯ection is given by the re¯ection coef®cient R

	R	jBj2	(2:53)
		jAj2

The wave function for the particle is obtained by joining the three parts øI, øII, and øIII such that the resulting wave function ø(x) and its ®rst derivative

ø9(x) are continuous. Thus, the following boundary conditions apply
øI(0)	øII(0),	ø9I(0)	ø9II(0)	(2:54)
øII(a)	øIII(a),	ø9II(a)	ø9III(a)	(2:55)

These four relations are suf®cient to determine any four of the constants A, B, C, D, F in terms of the ®fth. If the particle were con®ned to a ®nite region of space, then its wave function could be normalized, thereby determining the ®fth and ®nal constant. However, in this example, the position of the particle may range from ÿ1 to 1. Accordingly, the wave function cannot be normalized, the remaining constant cannot be evaluated, and only relative probabilities such as the transmission and re¯ection coef®cients can be determined.

We ®rst evaluate the transmission coef®cient T in equation (2.52). Applying equations (2.55) to (2.47 b) and (2.50), we obtain

Ceâa Deÿâa Feiáa

â(Ceâa ÿ Deÿâa) iáFeiáa from which it follows that

		2.6 Tunneling			55
C		F	â iá	e(iáÿâ)a
			2â
					(2:56)
			â ÿ iá
D		F		e(iá â)a
			2â

Application of equation (2.54) to (2.47 a) and (2.50) gives

A B C D

(2:57)

iá(A ÿ B) â(C ÿ D)

Elimination of B from the pair of equations (2.57) and substitution of equations (2.56) for C and for D yield

1

A 2iá [(â iá)C ÿ (â ÿ iá)D]

eiáa

F 4iáâ [(â iá)2eÿâa ÿ (â ÿ iá)2eâa]

At this point it is easier to form jAj2 before any further algebraic simpli®cations

jAj2 A A

jFj2					1		[(â2 á2)2eÿ2âa (â2 á2)2e2âa ÿ (â ÿ iá)4 ÿ (â iá)4]
			16á2â2
jFj2					1		[(á2 â2)2(eâa ÿ eÿâa)2 16á2â2]
			16á2â2
j	j		"				4á2	â2	#
	F	2	1			(á2		â2)2	sinh2 âa

where equation (A.46) has been used. Combining this result with equations (2.48), (2.51), and (2.52), we obtain

T "1	V	2	E) sinh2(		2m(V0 ÿ E) a=")#	ÿ1
	4E(V0	0				(2:58)
		ÿ			p••••••••••••••••••••••••

To ®nd the re¯ection coef®cient R, we eliminate A from the pair of equations (2.57) and substitute equations (2.56) for C and for D to obtain

B

1

[ (â

ÿ

iá)C

(â

iá)D]

F

eiáa

[(á2

â2)eâa

ÿ

(á2

â2)eÿâa]

2iá

4iáâ

ÿ

eiáa

F 2iáâ (á2 â2)sinh âa

56 SchroÈdinger wave mechanics

where again equation (A.46) has been used. Combining this result with equations (2.52) and (2.53), we ®nd that

R

T

jBj2

T

(á2 â2)2

sinh2 âa

4á2â2

jFj2

Substitution of equations (2.48), (2.51), and (2.58) yields

V 02

E)

2

2m(V0 ÿ E) a=")

4E(V

sinh (

p••••••••••••••••••••••••

(2:59)

R

0 ÿ2

V

0

E)

sinh2(

2m(V0 ÿ E) a=")

1

4E(V

p••••••••••••••••••••••••

0

ÿ

The transmission coef®cient T in equation (2.58) is the relative probability that a particle impinging on the potential barrier tunnels through the barrier. The re¯ection coef®cient R in equation (2.59) is the relative probability that the particle bounces off the barrier and moves in the negative x-direction. Since the particle must do one or the other of these two possibilities, the sum of T and R should equal unity

T R 1

which we observe from equations (2.58) and (2.59) to be the case.

We also note that the (relative) probability for the particle being in the region 0 < x < a is not zero. In this region, the potential energy is greater than the total particle energy, making the kinetic energy of the particle negative. This concept is contrary to classical theory and does not have a physical signi®- cance. For this reason we cannot observe the particle experimentally within the potential barrier. Further, we note that because the particle is not con®ned to a ®nite region, the boundary conditions on the wave function have not imposed any restrictions on the energy E. Thus, the energy in this example is not quantized.

In this analysis we considered the relative probabilities for tunneling and re¯ection for a single particle. The conclusions apply equally well to a beam of particles, each of mass m and total energy E, traveling initially in the positive x-direction. In that case, the transmission coef®cient T in equation (2.58) gives the fraction of incoming particles that tunnel through the barrier, and the re¯ection coef®cient R in equation (2.59) gives the fraction that are re¯ected by the barrier.

If the potential barrier is thick (a is large), the potential barrier is high compared with the particle energy E (V0 E), the mass m of the particle is large, or any combination of these characteristics, then we have

sinh âa	1	(eâa ÿ eÿâa)	eâa
	2		2

	2.7		Particles in three dimensions					57
so that T and R become		16E(V0 ÿ E) eÿ2ap2m(V0ÿE)="
T
							••••••••••••••••
R		1		V 02				ÿE)="
				16E(V0	ÿ E) eÿ2ap2m(V0
			ÿ				••••••••••••••••
				V02

In the limit as a ! 1, as V0 ! 1, as m ! 1, or any combination, the transmission coef®cient T approaches zero and the re¯ection coef®cient R approaches unity, which are the classical-mechanical values. We also note that in the limit " ! 0, the classical values for T and R are obtained.

Examples of tunneling in physical phenomena occur in the spontaneous emission of an alpha particle by a nucleus, oxidation±reduction reactions, electrode reactions, and the umbrella inversion of the ammonia molecule. For these cases, the potential is not as simple as the one used here, but must be selected to approximate as closely as possible the actual potential. However, the basic qualitative results of the treatment here serve to explain the general concept of tunneling.

2.7 Particles in three dimensions

Up to this point we have considered particle motion only in the x-direction. The generalization of SchroÈdinger wave mechanics to three dimensions is straightforward. In this section we summarize the basic ideas and equations of wave mechanics as expressed in their three-dimensional form.

The position of any point in three-dimensional cartesian space is denoted by

the vector r with components x, y, z, so that
r ix jy kz	(2:60)

where i, j, k are, respectively, the unit vectors along the x, y, z cartesian coordinate axes. The linear momentum p of a particle of mass m is given by

	dr		dx			dy		dz
p m			m i			j		k		i px j py k pz	(2:61)
	dt			dt			dt		dt
The x-component,			px, of the momentum now needs to carry a subscript,

whereas before it was denoted simply as p. The scalar or dot product of r and p is

r . p p . r xpx ypy zpz and the magnitude p of the vector p is

p (p . p)1=2 ( p2x p2y p2z )1=2

The classical Hamiltonian H(p, r) takes the form

58	SchroÈdinger wave mechanics
H(p, r)	p2		1	( p2x p2y p2z ) V (r)
		V(r)			(2:62)
	2m		2m
When expressed in three dimensions, the de Broglie relation is
		p "k			(2:63)

where k is the vector wave number with components kx, ky, kz. The de Broglie wavelength ë is still given by

ë 2kð hp

where now k and p are the magnitudes of the corresponding vectors. The wave packet representing a particle in three dimensions is

Ø(r, t)	1	… A(p, t)ei(p.rÿEt)=" dp	(2:64)
	(2ð")3=2

As shown by equations (B.19), (B.20), and (B.27), the momentum-space wave function A(p, t) is a generalized Fourier transform of Ø(r, t),

A(p, t)	1	…Ø(r, t)eÿi(p.rÿEt)=" dr	(2:65)
	(2ð")3=2

The volume elements dr and dp are de®ned as dr dx dy dz dp d px d py d pz

and the integrations extend over the complete range of each variable. For a particle moving in three-dimensional space, the quantity

Ø (r, t)Ø(r, t) dr Ø (x, y, z, t)Ø(x, y, z, t) dx d y dz

is the probability at time t of ®nding the particle with its x-coordinate between

x and x dx,	its y-coordinate between y and y	dy, and its z-coordinate
between z and	z dz. The product Ø (r, t)Ø(r,	t) is, then, the probability

density at the point r at time t. If the particle is under the in¯uence of an
external potential ®eld V(r), the wave function Ø(r, t) may be normalized
…Ø (r, t)Ø(r, t) dr 1	(2:66)

The quantum-mechanical operators corresponding to the components of p are

" @

" @

" @

^px

,

^py

,

^pz

i

@x

i

@ y

i

@z

or, in vector notation

"

:

p i =

^

(2 67)

2.7 Particles in three dimensions							59
where the gradient operator = is de®ned as
@		@			@
= i		j		k
	@x		@ y		@z

Using these relations, we may express the Hamiltonian operator in three dimensions as

2

H^

ÿ"

=2 V(r)

2m

where the laplacian operator =2 is de®ned by

=2

@2

@2

@2

= . =

@x2

@ y2

@z2

The time-dependent SchroÈdinger equation is

@Ø(r,

t)

^

i"

HØ(r, t)

@t

ÿ"2

=2Ø(r, t)

V (r)Ø(r, t)

(2:68)

2m

The stationary-state solutions to this differential equation are

Øn(r, t) øn(r)eÿiEn t="

(2:69)

where the spatial functions øn(r) are solutions of the time-independent SchroÈdinger equation

ÿ"2	=2ø	(r)		V (r)ø		(r)		E	ø	(r)	(2:70)
2m	n				n			n	n
The most general solution to equation (2.68) is
Ø(r, t) Xn					cnøn(r)eÿiEn t="						(2:71)

where cn are arbitrary complex constants.

The expectation value of a function f (r, p) of position and momentum is

given by
hf (r, p)i …Ø (r, t) f r, "i = Ø(r, t) dr	(2:72)

Equivalently, expectation values of three-dimensional dynamical quantities may be evaluated for each dimension and then combined, if appropriate, into vector notation. For example, the two Ehrenfest theorems in three dimensions are

60	SchroÈdinger wave mechanics
		p		m	dhri
		h i				dt
		dhpi	ÿh			=V	i h	F	i
		dt

where F is the vector force acting on the particle. The Heisenberg uncertainty principle becomes

	ÄxÄ px >	"	,	ÄyÄ py >	"	,	ÄzÄ pz >	"
		2			2			2

Multi-particle system

For a system of N distinguishable particles in three-dimensional space, the classical Hamiltonian is

	p2		p2		p2
H(p1, p2, . . . , pN , r1, r2, . . . , rN )	1		2		N
	2m1		2m2		2mN

V (r1, r2, . . . , rN )

where rk and pk are the position and momentum vectors of particle k. Thus, the quantum-mechanical Hamiltonian operator is

	"2		1			1			1
H^		ÿ			=12			=22			=2N V (r1, r2, . . . , rN ) (2:73)
		2		m1			m2			mN

where =2k is the laplacian with respect to the position of particle k.

The wave function for this system is a function of the N position vectors: Ø(r1, r2, . . . , rN , t). Thus, although the N particles are moving in threedimensional space, the wave function is 3N-dimensional. The physical interpretation of the wave function is analogous to that for the three-dimensional case. The quantity

Ø (r1, r2, . . . , rN , t)Ø(r1, r2, . . . , rN , t) dr1 dr2 . . . drN

Ø (x1, y1, z1, x2, . . . , zN )Ø(x1, y1, z1, x2, . . . , zN ) dx1 dy1 dz1 dx2 . . . dzN is the probability at time t that, simultaneously, particle 1 is between x1, y1, z1 and x1 dx1, y1 dy1, z1 dz1, particle 2 is between x2, y2, z2 and x2 dx2, y2 dy2, z2 dz2, . . . , and particle N is between xN , yN , zN and xN dxN,

yN dyN , zN dzN . The normalization condition is
…Ø (r1, r2, . . . , rN , t)Ø(r1, r2, . . . , rN , t) dr1 dr2 . . . drN 1	(2:74)

This discussion applies only to systems with distinguishable particles; for example, systems where each particle has a different mass. The treatment of wave functions for systems with indistinguishable particles is more compli-

2.8 Particle in a three-dimensional box

61

cated and is discussed in Chapter 8. Such systems include atoms or molecules with more than one electron, and molecules with two or more identical nuclei.

2.8 Particle in a three-dimensional box

A simple example of a three-dimensional system is a particle con®ned to a rectangular container with sides of lengths a, b, and c. Within the box there is no force acting on the particle, so that the potential V(r) is given by

V (r) 0,	0 < x < a, 0 < y < b, 0 < z < c
1,	x , 0, x . a; y , 0, y . b; z , 0, z . c

The wave function ø(r) outside the box vanishes because the potential is in®nite there. Inside the box, the wave function obeys the SchroÈdinger equation

(2.70) with the potential energy set equal to zero

!

ÿ"2

@2ø(r)

@2ø(r)

@2ø(r)

Eø(r)

(2:75)

2m

@x2

@ y2

@z2

The standard procedure for solving a partial differential equation of this type is to assume that the function ø(r) may be written as the product of three functions, one for each of the three variables

ø(r) ø(x, y, z) X (x)Y (y)Z(z)

(2:76)

Thus, X (x) is a function only of the variable x, Y (y) only of y, and Z(z) only of z. Substitution of equation (2.76) into (2.75) and division by the product XYZ give

	ÿ"2	d2 X		ÿ"2	d2 Y		ÿ"2	d2 Z		E	(2:77)
	2mX dx2			2mY d y2			2mZ dz2

The ®rst term on the left-hand side of equation (2.77) depends only on the variable x, the second only on y, and the third only on z. No matter what the values of x, or y, or z, the sum of these three terms is always equal to the same constant E. The only way that this condition can be met is for each of the three terms to equal some constant, say Ex, Ey, and Ez, respectively. The partial differential equation (2.77) can then be separated into three equations, one for

each variable

d2 X 2m

d2 Y 2m

d2 Z 2m

ExX 0,

EyY 0,

EzZ 0

dx2

"2

dy2

"2

dz2

"2

(2:78)

where

Ex Ey Ez E

(2:79)