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

62 SchroÈdinger wave mechanics

Thus, the three-dimensional problem has been reduced to three one-dimen- sional problems.

The differential equations (2.78) are identical in form to equation (2.34) and the boundary conditions are the same as before. Consequently, the solutions

inside the box are given by equation (2.40) as

X (x)

2

sin

nxðx

,

nx

1, 2, 3, . . .

a

a

r•••

Y (y)

2

sin

nyðy

,

ny

1, 2, 3, . . .

(2:80)

b

b

r•••

Z(z)

2

sin

nzðz

,

nz

1, 2, 3, . . .

c

c

r•••

and the constants Ex, Ey, Ez are given by equation (2.39)

Ex

n2x h2

,

nx 1, 2, 3, . . .

8ma2

Ey

n2y h2

,

ny 1, 2, 3, . . .

(2:81)

8mb2

n2 h2

Ez

z

,

nz 1, 2, 3, . . .

8mc2

The quantum numbers nx, ny, nz take on positive integer values independently

of each other. Combining equations (2.76) and (2.80) gives the wave functions
inside the three-dimensional box
	r•••			nxðx		nyðy		nzðz
ønx,n y,nz (r)	8		sin		sin		sin		(2:82)
		v						c
				a		b

where v abc is the volume of the box. The energy levels for the particle are obtained by substitution of equations (2.81) into (2.79)

h2

n2

n2y

n2

Enx,ny,nz

x

z

(2:83)

8m

a2

b2

c2

Degeneracy of energy levels

If the box is cubic, we have a b c and the energy levels become

h2

Enx,ny,nz

(n2x n2y n2z )

(2:84)

8ma2

The lowest or zero-point energy is E1,1,1 3h2=8ma2, which is three times the zero-point energy for a particle in a one-dimensional box of the same length. The second or next-highest value for the energy is obtained by setting one of

2.8 Particle in a three-dimensional box

63

Table 2.1. Energy levels for a particle in a threedimensional box with a b c

Energy		Degeneracy		Values of nx, ny, nz
3(h2/8ma2)		1	1,1,1
6(h2/8ma2)		3	2,1,1	1,2,1	1,1,2
9(h2	/8ma2)	3	2,2,1	2,1,2	1,2,2
11(h2	/8ma2)	3	3,1,1	1,3,1	1,1,3
12(h2	/8ma2)	1	2,2,2
14(h2	/8ma2)	6	3,2,1	3,1,2	2,3,1	2,1,3 1,3,2 1,2,3

Table 2.2. Energy levels for a particle in a threedimensional box with b a=2, c a=3

Energy	Degeneracy	Values of nx, ny, nz
14(h2/8ma2)	1	1,1,1
17(h2/8ma2)	1	2,1,1
22(h2/8ma2)	1	3,1,1
26(h2/8ma2)	1	1,2,1
29(h2/8ma2)	2	2,2,1	4,1,1
34(h2/8ma2)	1	3,2,1
38(h2/8ma2)	1	5,1,1
41(h2/8ma2)	2	1,1,2	4,2,1

the integers nx, ny, nz equal to 2 and the remaining ones equal to unity. Thus, there are three ways of obtaining the value 6h2=8ma2, namely, E2,1,1, E1,2,1, and E1,1,2. Each of these three possibilities corresponds to a different wave function, respectively, ø2,1,1(r), ø1,2,1(r), and ø1,1,2(r). An energy level that corresponds to more than one wave function is said to be degenerate. The second energy level in this case is threefold or triply degenerate. The zeropoint energy level is non-degenerate. The energies and degeneracies for the ®rst six energy levels are listed in Table 2.1.

The degeneracies of the energy levels in this example are the result of symmetry in the lengths of the sides of the box. If, instead of the box being cubic, the lengths of b and c in terms of a were b a=2, c a=3, then the values of the energy levels and their degeneracies are different, as shown in Table 2.2 for the lowest eight levels.

Degeneracy is discussed in more detail in Chapter 3.

64 SchroÈdinger wave mechanics

		Problems
2.1	Consider a particle in a one-dimensional box of length a and in quantum state n.
	What is the probability that	the particle is in the left quarter of the box
	(0 < x < a=4)? For which state n is the probability a maximum? What is the
	probability that the particle is in the left half of the box (0 < x < a=2)?
2.2	Consider a particle of mass m in a one-dimensional potential such that
	V(x) 0,	ÿa=2 < x < a=2
	1,	x , ÿa=2, x . a=2
	Solve the time-independent SchroÈdinger equation for this particle to obtain the
	energy levels and the normalized wave functions. (Note that the boundary
	conditions are different from those in Section 2.5.)

2.3Consider a particle of mass m con®ned to move on a circle of radius a. Express the Hamiltonian operator in plane polar coordinates and then determine the energy levels and wave functions.

2.4Consider a particle of mass m and energy E approaching from the left a potential

barrier of height V0, as shown in Figure 2.3 and discussed in Section 2.6. However, suppose now that E is greater than V0 (E . V0). Obtain expressions for the re¯ection and transmission coef®cients for this case. Show that T equals unity when E ÿ V0 n2ð2"2=2ma2 for n 1, 2, . . . Show that between these periodic maxima T has minima which lie progressively closer to unity as E increases.

2.5Find the expression for the transmission coef®cient T for Problem 2.4 when the energy E of the particle is equal to the potential barrier height V0.

3

General principles of quantum theory

3.1 Linear operators

The wave mechanics discussed in Chapter 2 is a linear theory. In order to develop the theory in a more formal manner, we need to discuss the properties

^
of linear operators. An operator A is a mathematical entity that transforms a
function ø into another function ö
ö	^	(3:1)
	Aø

Throughout this book a circum¯ex is used to denote operators. For example, multiplying the function ø(x) by the variable x to give a new function ö(x) may be regarded as operating on the function ø(x) with the operator ^x, where ^x means multiply by x: ö(x) ^xø(x) xø(x). Generally, when the operation is simple multiplication, the circum¯ex on the operator is omitted. The operator

^	ø(x) gives the ®rst derivative of			ø(x) with
Dx, de®ned as d=dx, acting on
respect to x, so that in this case
	^	dø
ö Dxø dx

^
The operator A may involve a more complex procedure, such as taking the
integral of ø with respect to x either implicitly or between a pair of limits.
^
The operator A is linear if it satis®es two criteria
^		^	^	(3:2a)
A(ø1	ø2) Aø1		Aø2
	^	^		(3:2b)
	A(cø) cAø

where c is any complex constant. In the three examples given above, the operators are linear. Some nonlinear operators are `exp' (take the exponential of) and [ ]2 (take the square of), since

65

66

General principles of quantum theory

ex y

exe y

ex

e y

6

e cx 6cex

[x y]2 x2 2xy y2 6x2 y2

The operator

^

[c(x y)]2 6c[x y]2

^

^

C is the sum of the operators A and

B if

^

^

^

^

^

Cø

(A B)ø Aø Bø

^

^

^

The operator C is the product of the operators

A and

B if

^

^^

^

^

Cø ABø

A(Bø)

^

ø and then

^

where ®rst B operates on

A operates on the resulting function.

Operators obey the associative law of multiplication, namely

^

^

^

^^

^

A(BC)

(AB)C

^2

of an operator

^

Operators may be combined. Thus, the square A

A is just the

^^

product AA

^2

^^

^ ^

A

ø AAø A(Aø)

^

Similar de®nitions apply to higher powers of A. As another example, the

differential equation

d2 y

k2 y 0

dx2

^ 2

2

^

2

2

) is the sum of

may be written as (Dx

k

)2y 0,2where the operator (Dx k

^

the two product operators Dx and k .

^

^^

^

In multiplication, the order of A and B is important because

ABø is not

^ ^

^

^

^

necessarily equal to BAø. For example, if

A

x and B

Dx, then we have

^^

^

while,

on

the

other

hand,

^ ^

^

ABø xDxø x(dø=dx)

BAø

Dx(xø)

ø x(dø=dx). The commutator of

^

^

^

^

A and

B, written as [A,

B], is an operator

de®ned as

^

^

^^

^ ^

(3:3)

[A,

B] AB

ÿ BA

^

^

^

^

^^

^

^

from which it follows that [A, B] ÿ[B, A]. If ABø BAø, then we have

^^

^ ^

^

^

0;

in

this

case

we

^

^

AB

BA and

[A,

B]

say that A and B commute. By

expansion of each side of the following expressions, we can readily prove the relationships

^ ^ ^	^ ^ ^	^ ^ ^	(3:4a)
[A, BC] [A, B]C		B[A, C]
^^ ^	^ ^ ^	^ ^ ^	(3:4b)
[AB, C] [A, C]B		A[B, C]

^	is the	^	^^	^ ^	1, where 1 may be
The operator A		reciprocal of B	if AB	BA

regarded as the unit operator, i.e., `multiply by unity'. We may write ^ ^ÿ1

A B

3.2 Eigenfunctions and eigenvalues

67

and ^ ^ÿ1. If the operator ^ possesses a reciprocal, it is , in

B A A non-singular

^		^		A^ÿ1ö. If
which case the expression ö		A^ø may be solved for ø, giving ø

A possesses no reciprocal, it is singular and the expression ö Aø may not be inverted.

3.2 Eigenfunctions and eigenvalues

Consider a ®nite set of functions f i and the relationship c1 f1 c2 f2 cn f n 0

where c1, c2, . . . are complex constants. If an equation of this form exists, then the functions are linearly dependent. However, if no such relationship exists, except for the trivial one with c1 c2 cn 0, then the functions are linearly independent. This de®nition can be extended to include an in®nite set of functions.

In general, the function ö obtained by the application of the operator ^ on

A an arbitrary function ø, as expressed in equation (3.1), is linearly independent of ø. However, for some particular function ø1, it is possible that

^	á1ø1
Aø1

where á1 is a complex number. In such a case ø1 is said to be an eigenfunction

^	and á1		^
of A		is the corresponding eigenvalue. For a given operator A, many
eigenfunctions may exist, so that
		^	(3:5)
		Aøi áiøi

where øi are the eigenfunctions, which may even be in®nite in number, and ái

are the corresponding eigenvalues. Each eigenfunction of ^ is unique, that is to

A

say, is linearly independent of the other eigenfunctions.

Sometimes two or more eigenfunctions have the same eigenvalue. In that situation the eigenvalue is said to be degenerate. When two, three, . . . , n eigenfunctions have the same eigenvalue, the eigenvalue is doubly, triply, . . . , n-fold degenerate. When an eigenvalue corresponds only to a single eigenfunc-

tion, the eigenvalue is non-degenerate.

A simple example of

an eigenvalue

equation involves the operator

^

Dx

mentioned in Section 3.1. When

^

kx

, the result is

Dx operates on e

^

kx

d

kx

ke

kx

Dxe

dx

e

Thus, the exponentials e

kx

are

eigenfunctions

of

^

Dx with corresponding

eigenvalues k. Since both the real part and the imaginary part of k can have any values from ÿ1 to 1, there are an in®nite number of eigenfunctions and these eigenfunctions form a continuum of functions.

ö (r, è, j)ø(r, è, j)r2 sin è dr dè dj

ö (x, y, z)ø(x, y, z) dx dy dz

68 General principles of quantum theory

	^	2
	Another example is the operator Dx acting on either sin nx or cos nx, where
	n is a positive integer (n > 1), for which we obtain
	^ 2	ÿn	2	sin nx
	Dx sin nx
	^ 2	ÿn	2	cos nx
	Dx cos nx				^ 2
	The functions sin nx and cos nx are eigenfunctions of					with eigenvalues
					Dx

ÿn2. Although there are an in®nite number of eigenfunctions in this example, these eigenfunctions form a discrete, rather than a continuous, set.

In order that the eigenfunctions øi have physical signi®cance in their application to quantum theory, they are chosen from a special class of functions, namely, those which are continuous, have continuous derivatives, are single-valued, and are square integrable. We refer to functions with these properties as well-behaved functions. Throughout this book we implicitly assume that all functions are well-behaved.

Scalar product and orthogonality

The scalar product of two functions ø(x) and ö(x) is de®ned as …1 ö (x)ø(x) dx

ÿ1

For functions of the three cartesian coordinates x, y, z, the scalar product of

ø(x, y, z) and ö(x, y, z) is

…1

ÿ1

For the functions ø(r, è, j) and ö(r, è, j) of the spherical coordinates r, è,

j, the scalar product is

…2ð…ð…1

0 0 0 „

In order to express equations in general terms, we adopt the notation dô to indicate integration over the full range of all the coordinates of the system

being considered and write the scalar product in the form

…

ö ø dô

For further convenience we also introduce a notation devised by Dirac and write the scalar product of ø and ö as hö…j øi, so that

hö j øi ö ø dô

The signi®cance of this notation is discussed in Section 3.6. From the de®nition of the scalar product and of the notation hö j øi, we note that

3.3 Hermitian operators

69

hö j øi hø j öi hö j cøi chö j øi hcö j øi c hö j øi

where c is an arbitrary complex constant. Since the integral hø j øi equals hø j øi, the scalar product hø j øi is real.

If the scalar product of ø and ö vanishes, i.e., if hö j øi 0, then ø and ö

are said to be	^	obey the
	orthogonal. If the eigenfunctions øi of an operator A
expressions
	høj j øii 0 all i, j with i 6 j

the functions øi form an orthogonal set. Furthermore, if the scalar product of øi with itself is unity, the function øi is said to be normalized. A set of functions which are both orthogonal to one another and normalized are said to be orthonormal

høj j øii äij		(3:6)
where äij is the Kronecker delta function,
äij 1,	i j	(3:7)
0,	i 6 j

3.3 Hermitian operators

		^
The linear operator A is hermitian with respect to the set of functions øi of the
variables q1	, q2	, . . . if it possesses the property that
		…øj A^øi dô …øi(A^øj) dô	(3:8)

The integration is over the entire range of all the variables. The differential dô has the form

dô w(q1, q2, . . .) dq1 dq2 . . .

where w(q1, q2, . . .) is a weighting function that depends on the choice of the

coordinates q1, q2, . . . For cartesian coordinates the weighting	function
w(x, y, z) equals unity; for spherical coordinates, w(r, è, j) equals	r2 sin è.

Special variables introduced to simplify speci®c problems have their own weighting functions, which may differ from unity (see for example Section 6.3). Equation (3.8) may also be expressed in Dirac notation

^	^	(3:9)
høj j Aøii hAøj j øii

in which the brackets indicate integration over all the variables using their weighting function.

70 General principles of quantum theory

For illustration, we consider some examples involving only one variable, namely, the cartesian coordinate x, for which w(x) 1. An operator that results in multiplying by a real function f (x) is hermitian, since in this case f (x) f (x) and equation (3.8) is an identity. Likewise, the momentum

operator

^p ("=i)(d=dx), which was introduced in Section 2.3, is hermitian

since

øj øi

1

" dø

"

1

"

1

dø

øj ^pøi dx

1 øj

i

dx

ÿ

øi

j

dx

…ÿ1

i

dx

i

i

…ÿ1

dx

…ÿ1

ÿ1

The integrated part is zero if the functions øi vanish

at in®nity, which they

øj

dx, so

must in order to be well-behaved. The remaining integral is

p

that we have

1

1

„

øi ^

…ÿ1øj ^pøi dx …ÿ1øi(^pøj) dx

The imaginary unit i contained in the operator ^p is essential for the

hermitian character of that operator. The operator ^ d=d is not hermitian

Dx x

because

1	dø	1	dø
…ÿ1øj	i	dx ÿ…ÿ1øi	j	dx	(3:10)
	dx		dx

where again the integrated part vanishes. The negative sign on the right-hand side of equation (3.10) indicates that the operator is not hermitian. The operator

^ 2, however, is hermitian.

Dx

The hermitian character of an operator depends not only on the operator itself, but also on the functions on which it acts and on the range of integration. An operator may be hermitian with respect to one set of functions, but not with respect to another set. It may be hermitian with respect to a set of functions de®ned over one range of variables, but not with respect to the same set over a different range. For example, the hermiticity of the momentum operator ^p is dependent on the vanishing of the functions øi at in®nity.

The product of two hermitian operators may or may not be hermitian.

	^^	^		^
Consider the product AB where A and				B are separately hermitian with respect
to a set of functions øi, so that
	^^		^	^		^ ^	(3:11)
	høj j ABøii hAøj			j Bøii hBAøj j øii
where we have assumed that the functions					^	^
					Aøi and Bøi also lie in the hermitian
^	^		^^				^	^
domain of A and	B. The product		AB is hermitian if, and only if,				A and	B
							^
commute. Using the same procedure, one can easily demonstrate that if A and
^				^^	^ ^	^ ^
B do not commute, then the operators (AB					BA) and i[A, B] are hermitian.
^	^			^^

By setting B equal to A in the product AB in equation (3.11), we see that the

square of a hermitian operator is hermitian. This result can be generalized to

3.3	Hermitian operators				71
^	^	2			^ ^
any integral power of A. Since jAøj			is always positive, the integral hAø j Aøi
is positive and consequently
			^2	øi > 0	(3:12)
	hø j A

Eigenvalues

The eigenvalues of a hermitian operator are real. To prove this statement, we consider the eigenvalue equation

	^	(3:13)
	Aø áø
^	^
where A is hermitian, ø is an eigenfunction of A, and á is the corresponding
eigenvalue. Multiplying by ø and integrating give
	^	(3:14)
	hø j Aøi áhø j øi

Multiplication of the complex conjugate of equation (3.13) by ø and integrating give

h ^ø j øi á hø j øi (3:15)

A

^
Because A is hermitian, the left-hand sides of equations (3.14) and (3.15) are
equal, so that
(á ÿ á )hø j øi 0	(3:16)

Since the integral in equation (3.16) is not equal to zero, we conclude that á á and thus á is real.

Orthogonality theorem

ø ø ^

If 1 and 2 are eigenfunctions of a hermitian operator A with different eigenvalues á1 and á2, then ø1 and ø2 are orthogonal. To prove this theorem, we begin with the integral

	^	(3:17)
hø2 j Aø1i á1hø2 j ø1i
^	is real, the left-hand side may be written as
Since A is hermitian and á2
^	^
hø2 j Aø1i hAø2 j ø1i á2hø2 j ø1i

Thus, equation (3.17) becomes

(á2 ÿ á1)hø2 j ø1i 0 Since á1 6 á2, the functions ø1 and ø2 are orthogonal.

Since the Dirac notation suppresses the variables involved in the integration,

we re-express the orthogonality relation in integral notation

…

ø2 (q1, q2, . . .)ø1(q1, q2, . . .)w(q1, q2, . . .) dq1 dq2 . . . 0

This expression serves as a reminder that, in general, the eigenfunctions of a