Physics 451 Quantum mechanics I Fall 2012 Oct 17, 2012 Karine Chesnel.

Physics 451

Quantum mechanics I

Fall 2012

Oct 17, 2012

Karine Chesnel

Next homework assignments:

•HW # 14 due Thursday Oct 18 by 7pm

Pb 3.7, 3.9, 3.10, 3.11, A26

• HW #15 due Tuesday Oct 23

Announcements

Phys 451

Practice test 2 M Oct 22 Sign for a problem!

Test 2 : Tu Oct 23 – Fri Oct 26

Quantum mechanics

Eigenvectors & eigenvalues

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

T a a

is transformed into a scalar multiple of itselfa

a is an eigenvector of T

is an eigenvalue of T

Quantum mechanics

Eigenvectors & eigenvalues

0T I a

det 0T I

To find the eigenvalues:

We get a Nth polynomial in : characteristic equation

Find the N roots 1 2, ,... N

Spectrum Pb A18, A25, A 26

Quantum mechanics

Gram-Schmidt Orthogonalization procedure

Discrete spectraDegenerate states

More than one eigenstate for the same eigenvalue

See problem A4, application A26

Quantum mechanics

Discrete spectra of eigenvalues

1. Theorem: the eigenvalues are real

2. Theorem: the eigenfunctions of distinct eigenvalues are orthogonal

3. Axiom: the eigenvectors of a Hermitian operator are complete

Quantum mechanics

Continuous spectra of eigenvalues

Q̂f x f x

No proof of theorem 1 and 2… but intuition for:

- Eigenvalues being real- Orthogonality between eigenstates- Compliteness of the eigenstates

Orthogonalization Pb 3.7

Quantum mechanics

Continuous spectra of eigenvalues

p p

df x pf x

i dx

Momentum operator:

For real eigenvalue p: - Dirac orthonormality

- Eigenfunctions are complete

Wave length – momentum: de Broglie formulae2

p

' ( ')p pf f p p

pf x c p f x dp

Quantum mechanics

Continuous spectra of eigenvalues

Position operator:

xf x f x

- Eigenvalue must be real

- Dirac orthonormality

- Eigenfunctions are complete

Quantum mechanics

Continuous spectra of eigenvalues

Eigenfunctions are not normalizableDo NOT belong to Hilbert spaceDo not represent physical states

If eigenvalues are real:- Dirac orthonormality- Eigenfunctions are complete

but

Generalized statistical interpretation

• Operator’s eigenstates: n n nQ q

eigenvectoreigenvalue

• Particle in a given state

• We measure an observable Q (Hermitian operator)

Eigenvectors are complete:

Discrete spectrum

1n n

n

c

Continuous spectrum

( ) ( )qc q x dq

Phys 451

Generalized statistical interpretation

Particle in a given state

• Normalization:

1n n

n

c

2

1n

n

c

• Expectation value

2

1n n

n

Q Q c q

Operator’s eigenstates: n n nQ q orthonormal

Phys 451

Quiz 18

A. the expectation value

B. one of the eigenvalues of Q

C. the average of all eigenvalues

D. A combination of eigenvalues

with their respective probabilities

If you measure an observable Q on a particle in a certain state ,

what result will you get?

Q

2

1n n

n

c q

1n

n

q

n nn

c

Phys 451

Operator ‘position’:

ˆy yxf x yf x

Generalized statistical interpretation

( ) ( ) ( , ) ( , )c y x y x t dx y t

Probability of finding the particle at x=y:2 2

( ) ( , )c y y t

Phys 451

p p

df x pf x

i dx

Operator ‘momentum’:

Generalized statistical interpretation

/1( ) ( , ) ,

2ipxc p e x t dx p t

Probability of measuring momentum p:2 2

( ) ( , )c p p t

px c p f x dp

Phys 451

Example Harmonic ocillator Pb 3.11

The Dirac notation

Different notations to express the wave function:

• Projection on the energy eigenstates

• Projection on the position eigenstates

• Projection on the momentum eigenstates

, ,x t y t x y dy

/

,2

ipxep t dp

/( ) niE tn n

n

c x e

Phys 451