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Physics 451

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Physics 451

Quantum mechanics I

Fall 2012

Oct 12, 2012

Karine Chesnel

Announcements

Homecoming

Quantum mechanics

Announcements

- Homework next week:
- HW # 13 due Tuesday Oct 16
- Pb 3.3, 3.5, A18, A19, A23, A25
- HW #14 due Thursday Oct 18
- Pb 3.7, 3.9, 3.10, 3.11, A26

Infinite- dimensional space

Wave function are normalized:

Hilbert space: functions f(x) such as

Quantum mechanics

Hilbert space

N-dimensional space

Wave functions live in Hilbert space

Norm

Orthonormality

Schwarz inequality

Quantum mechanics

Hilbert space

Inner product

Expectation value

since

For any f and g functions

Observables are Hermitian operators

Examples:

Quantum mechanics

Hermitian operators

Observable - operator

Stationary states – determinate energy

Generalization of

Determinate state:

Standard deviation:

For determinate state:

operator

eigenstate

eigenvalue

Quantum mechanics

Determinate states

Quantum mechanics

Quiz 16

Since any wave function can be written as a linear combination

of determinate states (stationary states), for which we can write

The wave function is itself a determinate state and we can write

- True
- B. False

For a given transformation T, there are “special” vectors for which:

is transformed into a scalar multiple of itself

is an eigenvector of T

l is an eigenvalue of T

Quantum mechanics

Eigenvectors & eigenvalues

Find the N roots

Spectrum

Quantum mechanics

Eigenvectors & eigenvalues

To find the eigenvalues:

We get a Nth polynomial in l: characteristic equation

Hermitian operator:

Quantum mechanics

Hermitian transformations

1. The eigenvalues are real

2. The eigenvectors corresponding to distinct eigenvalues are orthogonal

3. The eigenvectors span the space