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PHYS 30101 Quantum Mechanics

Lecture 14

Dr Gavin Smith Nuclear Physics Group

These slides at http//nuclear.ph.man.ac.uk/jb/p

hys30101

- Syllabus
- Basics of quantum mechanics (QM) Postulate,

operators, eigenvalues eigenfunctions,

orthogonality completeness, time-dependent

Schrödinger equation, probabilistic

interpretation, compatibility of observables,

the uncertainty principle. - 1-D QM Bound states, potential barriers,

tunnelling phenomena. - Orbital angular momentum Commutation relations,

eigenvalues of Lz and L2, explicit forms of Lz

and L2 in spherical polar coordinates, spherical

harmonics Yl,m. - Spin Noncommutativity of spin operators, ladder

operators, Dirac notation, Pauli spin matrices,

the Stern-Gerlach experiment. - Addition of angular momentum Total angular

momentum operators, eigenvalues and

eigenfunctions of Jz and J2. - The hydrogen atom revisited Spin-orbit coupling,

fine structure, Zeeman effect. - Perturbation theory First-order perturbation

theory for energy levels. - Conceptual problems The EPR paradox, Bells

inequalities.

4. Spin 4.1 Commutators, ladder operators,

eigenfunctions, eigenvalues 4.2 Dirac notation

(simple shorthand useful for spin space) 4.3

Matrix representations in QM Pauli spin

matrices 4.4 Measurement of angular momentum

components the Stern-Gerlach apparatus

RECAP 4. Spin (algebra almost identical to

orbital angular momentum algebra except we

cant write down explicit analogues of spherical

harmonics for spin eigenfunctions)

Commutation relations

(plus two others by cyclic permutation of x,y,z)

By convention we choose to work with

eigenfunctions of S2 and Sz which we label a and ß

So, the eigenvalue equations are

Any general spin-1/2 wavefunction ? can be

written as a linear combination of the complete

set of our chosen eigenfunction set

? a a b ß

(theres only two eigenfunctions in the set)

The coefficients a and b give the weighting and

relative phases of the a and ß eigenstates. Normal

ization a2 b2 1

The wavefunction ? could be, for example, that of

a spin-1/2 particle polarised in the x-direction

(an eigenstate of Sx) We now find the

coefficients a, b for this state as an example

Eigenfunctions and eigenvalues of Sx, Sy, Sz

described in this way

? a a b ß

RECAP 4.2 Dirac notation

Dirac

4.3 Matrix representations in QM

We can describe any function as a linear

combination of our chosen set of eigenfunctions

(our basis)

Substitute in the eigenvalue equation for a

general operator

Gives

4.3 Matrix representations in QM

We can describe any function as a linear

combination of our chosen set of eigenfunctions

(our basis)

Substitute in the eigenvalue equation for a

general operator

Equation (1)

Gives

Multiply from left and integrate

)

(We use

Exactly the rule for multiplying matrices!

And find

Matrix representation Eigenvectors of Sx, Sy, Sz

Eigenfunctions of spin operators (from lecture 13)