# 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)