Quantum Mechanics
1 The Wave Function
1.1 The Schrodinger Equation
Goal of Quantum Mechanics: Determine the wave function of the particle.
Schrodinger Equation:
Given the initial conditions: ; the Schrodinger equation determines
for all future time.
1.2 Probability
represents the probability of finding the particle between
and
.
– number of people with
:
– probability of getting
:
Expectation value (average):
Variance:
Standard deviation –
– probability that individual is between
and
– probability density
Continuous distributions:
1.3 Normalization
Particle must be somewhere:
Non-normalizable solutions cannot represent particles.
Wave function remains normalized as time goes on.
1.4 Momentum
Average of measurements performed on particles all in the state :
Through integration by parts and discarding boundary terms:
Expectation value of velocity:
Expectation value of momentum:
Notation with operators:
Expectation value of dynamical variables:
1.5 The Uncertainty Principle
de Broglie formula:
Heisenberg uncertainty principle:
2 The Time-Independent Schrodinger Equation
2.1 Stationary States
Separation of Variables:
Using the Schrodinger equation:
Crucial argument: Left side is a function of only , right side is a function of only
, therefore both sides are constant denoted by
.
Time-independent Schrodinger equation:
Separable Solutions
1. They are stationary states – the probability density does not depend on time.
- They are states of definite total energy.
Hamiltonian:
3. The general solution is a linear combination of separable solutions. There is a different wave function for each allowed energy.
Time-dependent Schrodinger equation has the property that any linear combination of solutions is itself a solution.
General solution:
2.2 The Infinite Square Well
if
, and
otherwise.
Outside the well:
Inside the well:
Simple harmonic oscillator:
General solution:
Boundary Conditions:
Hence:
Distinct solutions:
Normalize :
Solutions:
– ground state, other waves are excited states
Important properties of
- They are alternately even and odd (true when potential is an even function).
-
Each successive state has one more node (universal).
-
They are mutually orthogonal (quite general).
– Kronecker delta
- They are complete, in the sense that any other function,
, can be expressed as a linear combination of them
Expansion coefficients can be evaluated by Fourier’s trick:
Stationary States for Infinite Square Well:
General Solution:
2.3 The Harmonic Oscillator
Practically any potential is approximately parabolic. Virtually any oscillatory motion is approximately simple harmonic.
Solve:
2.3.1 Algebraic Method
Ladder Operators:
Schrodinger equation:
Important: If satisfies the Schrodinger equation with energy
, then
satisfies the Schrodinger equation with energy
.
is a solution with energy
.
There must exist a “lowest rung”:
Excited states:
2.3.2 Analytic Method
Substitution:
Schrodinger equation:
Large :
Asymptotic form:
Look for a solution in the form of a power series:
Recursion formula:
The power series must terminate for the solution to be normalizable. One series must truncate, the other must be zero from the start.
Stationary states:
2.4 The Free Particle
everywhere.
Special combination: represents a wave of fixed profile, traveling in the
direction, at speed
.
: Wave traveling to the right
: Wave traveling to the left
A free particle cannot exist in a stationary state; there is no such thing as a free particle with a definite energy.
Wave packet:
Plancherel’s theorem:
2.5 The Delta-Function Potential
Bound State: If rises higher than the particle’s total energy
on either side, then the particle is stuck in the potential well.
Scattering State: If exceeds
on one side or both, the the particle comes in from infinity and returns to infinity.
Tunneling allows the particle to leak through any finite potential barrier.
and
– Bound State
or
– Scattering State
Real life: Most potentials go to zero at infinity.
: Bound State
: Scattering State
Infinite square well and harmonic oscillator admit bound states only. Free particle only allows scattering states.
Dirac delta function
Potential:
2.5.1 Bound States:
General Solution:
Boundary Conditions:
1. is always continuous.
2. is continuous except at points where the potential is infinite.
Integrate the Schrodinger equation from to
and take the limit as
:
One bound state:
2.5.2 Scattering States:
In a typical scattering experiment particles are fired in from one direction. In that case the amplitude of the wave coming in from the right will be zero.
– amplitude of the incident wave
– amplitude of the transmitted wave
Reflection Coefficient
Transmission Coefficient
2.6 Finite Square Well
for
for
2.6.1 Bound States:
,
:
,
:
,
:
- Wide, deep well
2. Shallow, narrow well: There is always one bound state, no matter how weak the well becomes.
2.6.2 Scattering States:
:
:
(assuming there is no incoming wave in this region):
Energies for perfect transmission:
2.7 The Scattering Matrix
Arbitrary Localized Potentials
Region 1:
Region 2:
Region 3:
Scattering Matrix
Scattering from the left:
Scattering from the right:
If you want to locate the bound states, put in .
For more information see: Introduction to Quantum Mechanics (David J. Griffiths)
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