*Exploring Quantum Physics – Week 1 Lecture 2*

## Born Interpretation of the Schrödinger equation

**Born Rule**: is the probability of finding the quantum particle (described by ) in the volume at time *t*.

## Continuity Equation

Integrate over volume:

Recognizing Gauss’ theorem in the integral, we get:

From Born’s Rule, calculate the probability flux:

*Remember*

Apply to Schrödinger:

Expanding the Hamiltonian and canceling the potential gives:

*Remember K.E. is an operator:*

is the probability current

This connects the change in the probability of finding a particle in *V* with the flux of the probability current flowing through the surface.

## Quantum operators

**Classical properties (momentum, location, etc.) become operators acting on the wave function**

- momentum:
- location: (multiplication operator)
- kinetic energy:

E.g. Angular momentum becomes

## How do you measure an operator?

Consider measuring particle position…

*x* becomes a probability function *F(x)*

*Notation:*

This generalizes to:

For a property *X* with operator :

## Time-independent schrödinger

Recall the time-*dependent* Schrödinger equation:

Often, is time independent, so we can separate the variables:

Canceling the exponential gives us the **time-independent Schrödinger equation**:

Notice that this looks like an eigenvalue equation, where *E* is the eigen value and is the eigen vector.

## Hermitian operators

For any operator, , acting in a space of functions , one can define its **Hermitian-adjoint operator** with:

And is “Hermitian” if

Eigenvalue properties:

- eigenvalues (
*a*) are real if is Hermitian
- eigenvectors, , form a basis

## Physical Significance

- Physical observables in quantum mechanics are described by Hermitian (a.k.a. self-adjoint, though there are subtle differences we will ignore) operators,
- Eigenvalues of a physical operator determine possible values of the observable that actually can be measured in an experiment.

- Eigenvectors form a basis in the sense that a wave-function can be expressed as their linear combination.

## Super position principle in Q.M.

- If and are solutions to the Schrödinger equation,

,

Then is also a solution.
- This motivates the notion of a Hilbert space – a linear vector space where quantum states live.
- The wave function, , is a specific representation of a quantum state (much like coordinates of a vector).
**Dirac notation** for “vectors” of quantum states: and
- For a basis {}:

## How to choose a basis?

- Physical observables in quantum mechanics are associated with linear Hermitian operators.
- For a generic operator, , the eigenvalue problem defines eigenvectors that form a basis in the Hilbert space.
- is the wave-function in the
*a*-representation.
- Standard choices are:
- coordinate representation:
- momentum representation: