BioPHYSICAL chemistry

(10.40)

Two- and three-dimensional problems can be

solved by using the approach of separation of

variables. With this approach, the equation is

rewritten to separate the contributions from

each variable so that the partial differential equa-

tion becomes two full second-order equations.

Assume that the wavefunction can be written as

the product of two wavefunctions, each of which

is dependent only upon one variable:

ψ(x,y) =X(x)Y(y) (10.41)

We do not know a prioriwhether this can be used successfully; we will

simply try substituting eqn 10.41 into Schrödinger’s equations (eqn 10.40)

yielding:

(10.42)

To simplify this equation, we group the constants with the energy and

remove the wavefunction from the right-hand side by dividing by the

wavefunction. So dividing this expression by:

(10.43)

gives:

(10.44)

The right-hand term is a simple constant, independent of xor y. Since the

two left-hand terms are dependent upon different variables that are always

equal to a constant, then each must be equal to a constant. We can then set:

(10.45)

(10.46)

where Ex+Ey=E.

1 2

2

2

22

22

Yy^2

Yy

y

mE

m

Yy

y

y

()

d () ()

d

or

d

d

=− − =

Z

Z

EEY yy ()

12

2

2

22

22

Xx^2

Xx

x

mE

m

Xx

x

x

()

d () ()

d

or

d

d

=− − =

Z

Z

EEXxx ()

112

2

2

Xx^2

Xx

xYy

Yy

() y

()

()

d ()

d

d

d

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+

⎛

⎝

⎜⎜

⎞⎞

⎠

⎟⎟=−

2

2

mE

Z

−

21

2

m

Z XxYy()()

−+

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

Z^22

2

2

2 m Yy()x Xx() Xx()y^2 Yy()

d

d

d

d

==EX x Y y()()

−+

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

Z^22

2

2

2 mxxy y^2 xy E

∂

∂

ψ

∂

∂

() ψ(),,ψψ()xy,

208 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

Potential energy

x  0

y  0

V(x,y)  0

v  

y  Ly

x  Lx

Figure 10.8
A particle in a
two-dimensional box
with a potential of
zero inside the box
and an infinite
potential outside
the box.