QM Reminder

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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  s 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 . All information is contained in the wavefunction  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 n of the corresponding operator. Given a Hermetian Op with eigenvalues n and eigenvectors  n, the probability of measuring the eigenvalue n is

Postulates of QM If measurement of an observable gives a result n, then immediately afterward the system is in state  n. 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 Momentum Total Energy Angular Momentum r = ( x, y, z ) - Cartesian repn L = r x p - 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  x,t  = A sin(kx-  t) or A cos(kx-  t) or A e i(kx-  t)  is an eigenfunction of P x  is an eigenfunction of Tot E  is not an eigenfunction of position X

Eigenfns: Hydrogenic atom  nlm (r  )  is an eigenfunction of Tot E  is an eigenfunction of L 2 and L z  is an eigenfunction of parity units eV

Eigenfns: Hydrogenic atom  nlm (r  )  is not an eigenfn of position X, Y, Z  is not an eigenfn of the momentum vector P x, P y, P z  is not an eigenfn of L x and L y

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

Klein-Gordon Wave Equation Start with the relativistic constraint for free particle: E tot 2 – p 2 c 2 = m 2 c 4. [ E tot 2 – p 2 c 2 ]  (r,t) = m 2 c 4  (r,t). p 2 = p x 2 + p y 2 + p z 2  a Monster to solve

Dirac Wave Equation Wanted a linear relativistic equation [ E tot 2 – p 2 c 2  m 2 c 4 ]  (r,t) = 0 E tot 2 – p 2 c 2 = m 2 c 4 Change notation slightly p = ( p x, p y, p z ) ~ [P 4 2 c 2  m 2 c 4 ]  (r,t) = 0 P 4 = ( p o, ip x, ip y, ip z ) 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

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

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

What Good Wavefunctions Look Like ER 5-6

Sketching Pictures of Wavefunctions KE + V = E tot Prob ~   

Bad 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) Sketching Pictures of Wavefunctions KE

Re If E tot > V, then k Re  ~ kinda free particle Im If E tot < V, then k Im  ~ decaying exponential 2  /k ~ ~ wavelength  /k ~ 1/e distance KE + KE 

Sample  (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 DiffyQ to obtain solns Numerically –Do the DiffyQ integrations with code Creation-Annihilation Operators –Pattern matching techniques derived from 1D 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 (ER6.3-7) –Finite Square Well (ER App H)

Simple/Bare Coulomb Eigenfns: Bare Coulomb - stationary states  nlm (r  ) or R nl (r) Y lm (  )

Numerical Techniques Using expectations of what the wavefn should look like… –Numerical integration of 2 nd order DiffyQ –Relaxation methods –.. –Joe Blow’s idea –Willy Don’s idea –Cletus’ lame idea –.. ER 5.7, App G

SHO Creation-Annihilation Op Techniques Define: If you know the gnd state wavefn  o, 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