Particles moving in free space.
Consider a particle of mass m moving in free space, 1 dimension x, V (x) = 0 for all x.
H ( x) ( x)  E ( x)
  d


V
(
x
)
 2m dx
  ( x)  E ( x)
 d
 ( x)  E ( x)
2m dx
d
 2mE
 ( x) 
 ( x)
dx

2
2
2
2
2
2
2
2
2
a general solution for this equation is:
 ( x)  A exp(ikx)
where : k 
2mE
, as

2
d
 2mE
 ( x)  (ik) A exp(ikx)  k  ( x) 
 ( x)
dx

2
2
2
2
2
the energy E is the kinetic energy as V(x) = 0:
k 1
p
E 
 mv 
2m 2
2m
 p  k
2
2
2
2
There is no boundary condition; therefore there is no quantization of the energy. E can
have any value (>0).
The Normalization constant cannot be determined:
1   ( x) ( x) dx  A  exp(ikx)  exp(ikx) dx


**

**
2



 A  exp(ikx) exp(ikx) dx  A  dx  A (2)
2
2

2

the probability of the particle being at a point x is the same everywhere:
 ( x) ( x)dx  A exp(ikx)  exp(ikx)dx
**
2
**
 A dx
2
for all x
The function corresponds to motion in the +x direction. An equally good function is:
 ( x)  A exp(ikx)
which corresponds to motion in the –x direction.
As these functions are imaginary they cannot be represented by a real graph. They are
imaginary oscillating functions:
 ( x)  exp(ikx)  cos(kx)  i sin(kx)
and a plot of its real part shows it is a ‘wave’ function:
ψ(x)
π
0
2π
kx
λ
x
λ is the wavelength
when x = λ, k x = 2π
k  2 , k 
2

 2  h  h
 p  k     
   2  
which is the de Broglie relationship – one of the fundamental concepts of quantum
mechanics.
Does it matter that we assumed V(x) = 0?
Let V(x) ≠ 0 but is constant. How does this effect the wavelength of the wavefunction for
fixed energy E.
H ( x) ( x)  E ( x)
  d


V
(
x
)
 2m dx
  ( x)  E ( x)
V ( x)  V , a constant
 d
 ( x)  E  V  ( x)
2m dx
d
 2 m E  V 
 ( x) 
 ( x)
2
dx

2
2
2
2
2
2
2
2
 ( x)  expikx,
where k 

2m E  V  2

2


h
h

2 m E  V  p
λ gets longer as V gets larger or as (E – V ) gets smaller. (E – V ) = T, the kinetic energy.
In order to have quantized energies it is necessary to have motion restricted to a finite
space (a box, a circle, a sphere, a part of the x (or r) axis owing to a Potential Energy
function V(x) (or V(r)) as in the SHO (or H atom).
For a wall, V(x) = ∞ , x < 0 , V(x) = 0 , x ≥ 0
ψ(x) = 0 for x < 0 ; ψ(x) = A exp ( i k x) for x ≥ 0
V(x)
∞
0
x
However if V(x)  ∞ over a range of x, e.g. as the left hand side of the SHO:
x
Get a larger probability of the particle being in the region of rising V(x), and exponential
decay  0 as V(x)  ∞.