Lecture 9
Particle in a rectangular well
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus
Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the sponsoring agencies.
Motion in two or
more dimensions



The particle in a rectangular well extends the
previous 1D problem to 2D. This introduces
two important concepts:
Separation of variables – a very powerful
and general technique in reducing the
dimension of differential equations.
Degenerate eigenfunctions.
The particle in
a rectangular well
The Schrödinger
equation for this is:
2
æ ¶2
¶2 ö
+ 2 ÷ Y = EY
2
ç
2m è ¶x ¶y ø
 Boundary conditions
are:

 ( x , y )  0, x  0, L1  x
y  0, L 2  y
Separation of variables


When a differential equation is two or higher
dimensional such as
2
æ ¶2
¶2 ö
+ 2 ÷ Y = EY
2
ç
2m è ¶x ¶y ø
We must always attempt separation of
variables. With this, a 2D problem breaks
down into two 1D problems. This happens if
the solution is the product of functions of
each of the variables Ψ = X(x)Y(y) with no
cross term like Z(x,y).
Separation of variables

To see separation of variables indeed occurs,
we first assume it does and write the solution
in the product form:
æ ¶2
¶2 ö
+ 2 ÷ Y(x, y) = EY
2
ç
2m è ¶x ¶y ø
2
æ ¶2
¶2 ö
+ 2 ÷ X (x)Y ( y) = EX (x)Y ( y)
2
ç
2m è ¶x ¶y ø
2
Separation of variables

The partial derivative ∂2/∂x2 will act only on
the X(x) part (similarly for ∂2/∂y2 on Y), hence
2
æ ¶2
æ ¶2 X
¶2 ö
¶2Y ö
+ 2 ÷ X (x)Y ( y) = Y + X 2 ÷ = EXY
2
2
ç
ç
2m è ¶x ¶y ø
2m è ¶x
¶y ø
2

Divide by XY the both sides.
æ 1 ¶2 X 1 ¶ 2Y ö
+
=E
2
2÷
ç
2m è X ¶x
Y ¶y ø
2

It has the form: f(x) + g(y) = e. This
immediately means f(x) and g(y) are
individually constant. Separation of variable
indeed took place.
F(x) + G(y) = constant
Separation of variables
æ 1 ¶2 X 1 ¶2Y ö
+
= E = E X + EY
2
2÷
ç
2m è X ¶x
Y ¶y ø
2
1 ¶2 X
= EX
2
2m X ¶x
1 ¶2Y
= EY
2
2m Y ¶y
¶ X
= EX X
2
2m ¶x
¶2Y
= EY Y
2
2m ¶y
2
2
2
2
2
These are the particle in a box equations!
Separation of variables
 2 
X n1 ( x )  

L
 1
1/ 2
sin
L1
2
E n1 
 2 
Yn2 ( y )  

L
 2 
n1 x
n1 h
sin
E n2 
2
1
8m L
 ( x , y )  X n1 ( x )Y n 2 ( y ) 
2
1
2
 L1 L 2 
n h
2
2
1
8m L

1/ 2
2
2
sin
n h
n2 h
2
8 m L2
2
2
8 m L2
n1 x
L1
2
n 2 y
L2
2
2
E n1 , n 2  E n1  E n 2 
1/ 2
sin
n 2 y
L2
0  x  L1 ; 0  y  L 2
n1 = 1,2,… ;n2 = 1,2,…
The particle in a
three-dimensional box

The argument can be easily extended to 3D:


8
 ( x, y, z )  

 L1 L 2 L 3 
1/ 2
sin
n1 x
sin
n 2 y
L1
sin
n 3 z
L2
L3
0  x  L1 ; 0  y  L 2 ; 0  z  L3
2
E n1 , n 2 , n3  E n1  E n 2  E n3 
n1 h
2
2
1
8m L
2

n2 h
2
2
8m L
2
2

n3 h
2
8m L
We now have three quantum numbers.
2
3
Degeneracy

Let us suppose L1 = L2 = L in the 2D case.
Then the energy is,
2
E n1 , n 2 

n1 h
2
8m L
2
2

n2 h
2
8m L
 ( n1  n 2 )
2
2
2
h
2
8m L
2
This expression gives identical energy for (n51h, 2
n2) = (2,1) or (1,2). We say the energy
is 2
8m L
doubly degenerate in that two different
eigenfunctions correspond to this eigenvalue.
Degeneracy


The degeneracy is often
caused by high symmetry.
For the square well case,
(n1, n2) = (2,1) and (1,2)
wave functions are related
by 90° rotation around
the center.
These two states are
distinguished by different
probability densities.
90° rotation
90° rotation
Degeneracy

When two wave functions Ψ1 and Ψ2 are
doubly degenerate:
Hˆ   E  an d Hˆ   E 
1


1
2
2
Then any linear combination of Ψ1 and Ψ2 is
also an eigenfunction with the same
eigenvalue.
Hˆ  c1  1  c 2  2   E  c1  1  c 2  2 
We can use this property to make Ψ1 and Ψ2
orthogonal to each other.
Summary


We have introduced the powerful separation
of variables technique for differential
equations in two or higher dimensions. All
two- and higher-dimensional Schrödinger
equations we study in this course depends on
this powerful technique.
Some eigenvalues are degenerate – more
than one eigenfunctions correspond to one
eigenvalue.