QM Reminder C Nave gsu edu http hyperphysics

QM Reminder

C Nave @ gsu. edu http: //hyperphysics. phy-astr. gsu. edu/hbase/quacon. html#quacon

Outline • Postulates of QM • Picking Information Out of Wavefunctions – Expectation Values – Eigenfunctions & Eigenvalues • Where do we get wavefunctions from? – Non-Relativistic – Relativistic • What good-looking Ys look like • Techniques for solving the Schro Eqn – Analytically – Numerically – Creation-Annihilation Ops

Postulates of Quantum Mechanics • The state of a physical system is completely described by a wavefunction Y. • All information is contained in the wavefunction Y • Probabilities are determined by the overlap of wavefunctions

Postulates of QM • Every measurable physical quantity has a corresponding operator. • The results of any individ measurement yields one of the eigenvalues ln of the corresponding operator. • Given a Hermetian Op with eigenvalues ln and eigenvectors Fn , the probability of measuring the eigenvalue ln is

Postulates of QM • If measurement of an observable gives a result ln , then immediately afterward the system is in state fn. • The time evolution of a system is given by • . corresponds to classical Hamiltonian

Picking Information out of Wavefunctions Expectation Values Eigenvalue Problems

Common Operators • Position r = ( x, y, z ) - Cartesian repn • Momentum • Total Energy • Angular Momentum L=rxp - work it out

Using Operators: A • Usual situation: Expectation Values • Special situations: Eigenvalue Problems the original wavefn a constant (as far as A is concerned)

Expectation Values • Probability Density at r • Prob of finding the system in a region d 3 r about r • Prob of finding the system anywhere

• Average value of position r • Average value of momentum p • Expectation value of total energy

Eigenvalue Problems Sometimes a function fn has a special property eigenvalue eigenfn Since this is simpler than doing integrals, we usually label QM systems by their list of eigenvalues (aka quantum numbers).

Eigenfns: 1 -D Plane Wave moving in +x direction Y(x, t) = A sin(kx-wt) or A cos(kx-wt) or A ei(kx-wt) • Y is an eigenfunction of Px • Y is an eigenfunction of Tot E • Y is not an eigenfunction of position X

Eigenfns: Hydrogenic atom Ynlm(r, q, f) • Y is an eigenfunction of Tot E • Y is an eigenfunction of L 2 and Lz • Y is an eigenfunction of parity units e. V

Eigenfns: Hydrogenic atom Ynlm(r, q, f) • Y is not an eigenfn of position X, Y, Z • Y is not an eigenfn of the momentum vector Px , Py , Pz • Y is not an eigenfn of Lx and Ly

Where Wavefunctions come from

Where do we get the wavefunctions from? • Physics tools – Newton’s equation of motion – Conservation of Energy – Cons of Momentum & Ang Momentum The most powerful and easy to use technique is Cons NRG.

Schrödinger Wave Equation Use non-relativistic formula for Total Energy Ops and http: //www-groups. dcs. st-and. ac. uk/~history/Mathematicians

Klein-Gordon Wave Equation Start with the relativistic constraint for free particle: Etot 2 – p 2 c 2 = m 2 c 4. p 2 = px 2 + py 2 + pz 2 [ Etot 2 – p 2 c 2 ] Y(r, t) = m 2 c 4 Y(r, t). a Monster to solve

Dirac Wave Equation Wanted a linear relativistic equation Etot 2 – p 2 c 2 = m 2 c 4 p = ( px , py , pz ) [ Etot 2 – p 2 c 2 - m 2 c 4 ] Y(r, t) = 0 Change notation slightly ~ P 4 = ( po , ipx , ipy , ipz ) [P 42 c 2 - m 2 c 4 ] Y(r, t) = 0 difference of squares can be factored ~ ( P 4 c + mc 2) (P 4 c-mc 2) and there are two options for how to do overall +/- signs 4 coupled equations to solve.

Time Dependent Schro Eqn Where H = KE + Potl E

ER 5 -5 Time Dependent Schro Eqn Where H = KE + Potl E

Time Independent Schro Eqn KE involves spatial derivatives only If Pot’l E not time dependent, then Schro Eqn separable ref: Griffiths 2. 1

Drop to 1 -D for ease

ER 5 -6 What Good Wavefunctions Look Like

Sketching Pictures of Wavefunctions Prob ~ Y* Y KE + V = Etot

Bad Wavefunctions

Sketching Pictures of Wavefunctions To examine general behavior of wave fns, look for soln of the form where k is not necessarily a constant (but let’s pretend it is for a sec) KE

KE - KE + If Etot > V, then k Re If Etot < V, then k Im Y ~ kinda free particle Y ~ decaying exponential 2 p/k ~ l ~ wavelength 1/k ~ 1/e distance

Sample Y(x) Sketches • • Free Particles Step Potentials Barriers Wells

Free Particle Energy axis V(x)=0 everywhere

1 -D Step Potential

1 -D Finite Square Well

1 -D Harmonic Oscillator

1 -D Infinite Square Well

1 -D Barrier

NH 3 Molecule

E&R Ch 5 Prob 23 Discrete or Continuous Excitation Spectrum ?

E&R Ch 5, Prob 30 Which well goes with wfn ?

Techniques for solving the Schro Eqn. • Analytically – Solve the Diffy. Q to obtain solns • Numerically – Do the Diffy. Q integrations with code • Creation-Annihilation Operators – Pattern matching techniques derived from 1 D SHO.

Analytic Techniques • Simple Cases – Free particle (ER 6. 2) – Infinite square well (ER 6. 8) • Continuous Potentials – 1 -D Simple Harmonic Oscillator (ER 6. 9, Table 6. 1, and App I) – 3 -D Attractive Coulomb (ER 7. 2 -6, Table 7. 2) – 3 -D Simple Harmonic Oscillator • Discontinuous Potentials – Step Functions (ER 6. 3 -7) – Barriers (ER 6. 3 -7) – Finite Square Well (ER App H)

Eigenfns: Bare Coulomb - stationary states Ynlm(r, q, f) or Rnl(r) Ylm(q, f) Simple/Bare Coulomb

Numerical Techniques ER 5. 7, App G • Using expectations of what the wavefn should look like… – – – – – Numerical integration of 2 nd order Diffy. Q Relaxation methods. . Joe Blow’s idea Willy Don’s idea Cletus’ lame idea. .

SHO Creation-Annihilation Op Techniques Define: If you know the gnd state wavefn Yo, then the nth excited state is:

Inadequacy of Techniques • Modern measurements require greater accuracy in model predictions. – Analytic – Numerical – Creation-Annihilation (SHO, Coul) • More Refined Potential Energy Fn: V() – Time-Independent Perturbation Theory • Changes in the System with Time – Time-Dependent Perturbation Theory