Download Lecture 25: Wave mechanics

Lecture 25: Wave mechanics
Review
Notions of quantum mechanics
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Energy is quantized
Light can be thought of as a particle
Electron can be thought of as a Wave
Wave-particle duality
Location of a particle, how certain can we be?
Today
o Heisenberg’s Uncertainty principle
o Waves and their mathematical description
o Wave equation: Harmonic oscillator
o Application to quantum mechanics
o Concept of operators
o Schrodinger’s equation
o Particle in the Box problem
Heisenberg’s Uncertainty principle
Heseinberg, a student of Niels Bohr, recognized the
true consequence of wave-particle duality. He reasoned that
if we were to describe position and momentum of
“quantum size” particle it will be difficult to measure both
the quantities simultaneously. That is, the act of measuring
position of a particle, say by shining a light on it, will
influence the momentum of the particle creating
uncertainty in both the quantities. He showed that:
x.p 
h
4
The implications of the simple equation are profound.
It took away the infinite certainty implicit in the Newton’s
description of motion and position of a particle. That is, in
classical mechanics, we can “measure” the position and
location of the particle with any degree of certainty using
ever increasing sophisticated techniques.
When Einstein learned about the Heisenberg’s theory,
he was deeply saddened. He is claimed to have said “ God
does not play dice with Men.” However, the profound truth
of the above equation endured despite Einstein and other
physicists attempts to find some flaw in the reasoning that
lead to the above equation.
Since the quantum particle cannot located with
arbitrary precision, we must describe it in terms of it’s
probability of finding at a given position.
Wave mechanics
Thanks to De Broglie, we know that particles can act
like waves. So it is interesting review some basic concepts
about waves. Consider an example of rolling hills; we may
sketch them as follows.
1.0
Height
0.5
0.0
-0.5
-1.0
-10
-5
0
5
10
Distance
Note how the structure is periodic. If we want to
describe topological height as a function of distance, we
can employ a simple trigonometric sine function:
Height  sin( Dis tan ce)
If we want to change the periodicity we can simply
substitute for variable Distance = k.Distance. As we increase
the magnitude of k, the period between the oscillation
decreases and vice versa. The periodicity is best described by
the wavelength, =2/k. So higher the k lower is the periodicity.
So for a space dependent wave we may write:
 2 
y  A sin( kx)  A sin 
x
  
A is the amplitude, and  is the period of the wave. Just as
wave can occur in space, there can be many properties that can
have a wave-like behavior in time, for example sound waves.
Harmonic Oscillator
Consider a small weight attached to a mass-less spring. If
we pull the weight, stretching the spring, and release it, the
attached mass will compress and expand the spring following a
periodic motion. How do we describe the motion of the mass in
this case? We use the Newton’s law:
dp
dv
d 2x
F  ma 
m m
  kx
dt
dt
dt
d 2x
k
 x
dt
m
x  A.sin( at )
dx
 v  Aa cos( at )
dt
dv d 2 x

  Aa 2 sin( at )  a 2 x
dt
dt
 a2  k / m
Thus motion of a harmonic oscillator is sinusoidal in time.
Comparing this equation with the space dependent variation
equation:
 2 
y  A sin( kx)  A sin 
x
  
 2 
x  A sin( at )  A sin 
t



where a=2/=k/m=oscillation frequency(rad/sec) and
1/=(frequency,Hz)
Notion of standing waves
Consider a string whose both ends are attached to solid
support, like a guitar. If we stretch it and release it, then only
certain wavelengths are allowed. This is because at the tied
ends the amplitude waves must be zero.
Thus notion of Quantization of wavelength arises, even naturally in
classical problems. This was exploited by Schrodinger develop a
famous equation that bears his name.
Wave mechanics
As we have seen, de Broglie suggested that any particle has
wavelength given by =h/mv. Thus, electron could also be
treated as wave. For example consider following problem.
Thus the electron wavelength is comparable to typical
molecular bond length! To describe the motion as well as
location of this electron wave we must develop an equation like
Newton’s force =ma equation. How can we use Bohr’s picture
to define the electron as a wave in it’s own orbit?
Electron as a standing wave in Orbit
Schrodinger equation
Schrodinger began developing the theory of electron waves
with a classical general time and space dependent equation of a
wave.
where c is the velocity of the wave.  is a function that
depends on both time and distance. The solution of this
equation can be written as:
Note here we have 2v=a, and De Moivre’s theorem identity
gives us:
e i  cos( )  i sin(  )
Thus the solutions (wavefunctions) are sums of sine and cosine
functions but for algebric reason it is useful to maintain
complex exponential notation. The key insight is to recognize
we can substitute for electron wavelength,=h/p, and for
frequency =E/h
Schrodinger equation (continued)
Similarly we can show by differentiating with respect to
position.
Note the last equation defines entirely new concept; the
momentum is an operator. He made use of this connection to
describe the total energy of electron system, which is simply
sum of potential and kinetic energy.
Whence, we can write
Hˆ   E
 h 2  2
 
2
 2m x
 ˆ
  V  E
