1 The Wave Function
1.1 The Schrodinger Equation
Goal of Quantum Mechanics: Determine the wave function of the particle.
Given the initial conditions: ; the Schrodinger equation determines for all future time.
represents the probability of finding the particle between and .
– number of people with :
– probability of getting :
Expectation value (average):
Standard deviation –
– probability that individual is between and
– probability density
Particle must be somewhere:
Non-normalizable solutions cannot represent particles.
Wave function remains normalized as time goes on.
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:
1. They are stationary states – the probability density does not depend on time.
- They are states of definite total energy.
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.
2.2 The Infinite Square Well
if , and otherwise.
Outside the well:
Inside the well:
Simple harmonic oscillator:
– 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:
2.3 The Harmonic Oscillator
Practically any potential is approximately parabolic. Virtually any oscillatory motion is approximately simple harmonic.
2.3.1 Algebraic Method
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”:
2.3.2 Analytic Method
Look for a solution in the form of a power series:
The power series must terminate for the solution to be normalizable. One series must truncate, the other must be zero from the start.
2.4 The Free Particle
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.
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
2.5.1 Bound States:
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
2.6 Finite Square Well
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
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)