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Chapter 11   Lecture 10 - The Infinite Square Well

The one dimensional Schrödinger equation is
-
2
2m
2
x2
Y (x)+V(x)Y (x)=EY (x)

11.1   Free Particle

A free particle is where V(x)=0
-
2
2m
2
x2
Y (x)=EY (x)     (11.1)
since E is only kinetic
E=
p2
2m
Þ
2
x2
Y (x)=- æ
ç
ç
è
p
 
ö
÷
÷
ø
2



 
Y (x)
The general solutions of this are
Y (x)=C1e
p
 
x
 
+C2e
-
p
 
x
 
=C1'sin
æ
ç
ç
è
p
 
x ö
÷
÷
ø
+C2'cos
æ
ç
ç
è
p
 
x ö
÷
÷
ø
    (11.2)
Features of our solutions

11.2   Infinite Square Well


Figure 10.1 - An Infinite Square Well
Where V=¥ (regions I and III) to remain finite the Schrödinger equation only permits Y (x)=0 . Where V=0 (region II) the Schrödinger equation becomes
-
2
2m
2
x2
Y (x)=EY (x)
and so the satisfactory solutions for this are of the form
Y (x)=Acoskx+Bsinkx     (11.3)
where
k=
p
 
= æ
ç
ç
è
2mE
2
ö
÷
÷
ø
1
2



 
    (11.4)
But the constraints placed in matching Y (x) in regions I, II and III are known as the boundary conditions. These are as follows
  1. Y (x) should be continuous and single valued
  2. limx® ¥Y (x)® 0 (ensures normalisation)
  3. dY (x)/dx should also be continuous if V(x) is finite
For an infinite square well the second boundary condition is already satisfied since
Y (x)=0,  |x|>a
From the first boundary condition Y (x) at the boundaries between the regions must match
x=a(Acos(ka)+Bsin(ka))=0 x=-a(Acos(ka)-Bsin(ka))=0     (11.5)
This boundary condition can be satisfied in two ways
  1. B=0 therefore cos(ka)=0 , this provides the cosine solutions.
    ka=
    np
    2
    		  (n=odd integers) k=
    np
    2a
    		    (11.6)
  2. A=0 therefore sin(ka)=0 , this provides the sine soltions.
    ka=
    np
    2
    		  (n=even integers) k=
    np
    2a
    		    (11.7)
Properties of these boundary conditions however as p= k we can obtain the equation
En=
2k2
2m
=
2p2n2
8ma2
    (11.9)

Normalisation
Y (x)=C é
ê
ê
ë
cos
 
sin
ù
ú
ú
û		 æ
ç
ç
è
np
2a
x ö
÷
÷
ø
C2 ó
õ
a


-a
sin2 æ
ç
ç
è
np
2a
x ö
÷
÷
ø		 dx=1 C2 ó
õ
a


-a
cos2 æ
ç
ç
è
np
2a
x ö
÷
÷
ø		 dx=1     (11.10)
Þ C=
1
a
    (11.11)

Figure 10.2 - Wave Functions in the Infinite Square Well

11.2.1   Probability Density

|Y (x)|2=
1
a
é
ê
ê
ë
cos
 
sin
ù
ú
ú
û
2



 
æ
ç
ç
è
np
2a
x ö
÷
÷
ø		     (11.12)

11.2.2   Energy Scales

En=
2p2n2
8ma2
  (for n=1)

Nuclear Scale

a=2× 10-15m
mp=1.67× 10-27kg
Þ E1~ 11.5MeV

Atomic Scale

a=10-10m
me=9.1× 10-31kg
Þ E1~ 8.5eV

Quantum Well Structure Scale

a=10-9m
mp=9.1× 10-31kg
Þ E1~ 0.085eV

Snooker Scale

a=1m
m=1kg
Þ E1~ 10-67J

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