Developing Wave Equations

1. Developing Wave Equations • Need wave equation and wave function for particles. Schrodinger, Klein-Gordon, Dirac • not derived. Instead forms were guessed at, then solved, and found where applicable • So Dirac equation applicable for spin 1/2 relativistic particles • Start from 1924 DeBroglie hypothesis: “particles” (those with mass as photon also a particle…) have wavelength l = h/p P460 - dev. wave eqn.

2. Wave Functions • Particle wave functions are similar to amplitudes for EM waves…gives interference (which was used to discover wave properties of electrons) • probability to observe =|wave amplitude|2=|y(x,t)|2 • particles are now described by wave packets • if y = A+B then |y|2 = |A|2 + |B|2 + AB* + A*B giving interference. Also leads to indistinguishibility of identical particles t1 t2 merge vel=<x(t2)>-<x(t1)> (t2-t1) Can’t tell apart P460 - dev. wave eqn.

3. Wave Functions • Describe particles with wave functions • y(x) = S ansin(knx) Fourier series (for example) • Fourier transforms go from x-space to k-space where k=wave number= 2p/l. Or p=hbar*k and Fourier transforms go from x-space to p-space • position space and l/k/momentum space are conjugate • the spatial function implies “something” about the function in momentum space P460 - dev. wave eqn.

4. Wave Functions (time) • If a wave is moving in the x-direction (or -x) with wave number k can have • kx-wt = constant gives motion of wave packet • the sin/cos often used for a bound state while the exponential for a right or left traveling wave P460 - dev. wave eqn.

5. Wave Functions (time) • Can redo Transform from wave number space (momentum space) to position space • normalization factors 2p float around in Fourier transforms • the A(k) are the amplitudes and their squares give the relative probability to have wavenumber k (think of Fourier series) • could be A(k,t) though mostly not in our book • as different k have different velocities, such a wave packet will disperse in time. See sect. 2-2. Not really 460 concern….. P460 - dev. wave eqn.

6. Heisenberg Uncertainty Relationships • Momentum and position are conjugate. The uncertainty on one (a “measurement”) is related to the uncertainty on the other. Can’t determine both at once with 0 errors • p = hbar k • electrons confined to nucleus. What is maximum kinetic energy? Dx = 10 fm • Dpx = hbarc/(2c Dx) = 197 MeV*fm/(2c*10 fm) = 10 MeV/c • while <px> = 0 • Ee=sqrt(p*p+m*m) =sqrt(10*10+.5*.5) = 10 MeV electron can’t be confined (levels~1 MeV) proton Kp = .05 MeV….can be confined P460 - dev. wave eqn.

7. Heisenberg Uncertainty Relationships • Time and frequency are also conjugate. As E=hf leads to another “uncertainty” relation • atom in an excited state with lifetime t = 10-8 s • |y(t)|2 = e-t/t as probability decreases • y(t) = e-t/2teiMt (see later that M = Mass/energy) • Dt ~ t DE = hDn Dn > 1/(4p10-8) > 8*106 s-1 • Dn is called the “width” or • and can be used to determine ths mass of quickly decaying particles • if stable system no interactions/transitions/decays P460 - dev. wave eqn.

8. Schrodinger Wave Equation • Schrodinger equation is the first (and easiest) • works for non-relativistic spin-less particles (spin added ad-hoc) • guess at form: conserve energy, well-behaved, predictive, consistent with l=h/p • free particle waves P460 - dev. wave eqn.

9. Schrodinger Wave Equation • kinetic + potential = “total” energy K + U = E • with operator form for momentum and K gives (Hamiltonian) Giving 1D time-dependent SE For 3D: P460 - dev. wave eqn.

10. Operators (in Ch 3) • Operators transform one function to another. Some operators have eigenvalues and eigenfunctions Only some functions are eigenfunctions. Only some values are eigenvalues In x-space or t-space let p or E be represented by the operator whose eigenvalues are p or E P460 - dev. wave eqn.

11. Operators • Hermitian operators have real eigenvalues and can be diagonalized by a unitary transformation • easy to see/prove for matrices Continuous function look at “matrix” elements P460 - dev. wave eqn.

12. Operators By parts • Example 1 O = d/dx Usually need function to be well-behaved at boundary (in this case infinity). P460 - dev. wave eqn.

13. Commuting Operators • Some operators commute, some don’t (Abelian and non-Abelian) • if commute [O,P]=0 then can both be diagonalize (have same eigenfunction) • conjugate quantities (e.g. position and momentum) can’t be both diagonalized (same as Heisenberg uncertainty) (sometimes) P460 - dev. wave eqn.

14. Interpret wave function as probability amplitude for being in interval dx P460 - dev. wave eqn.

15. Example No forces. V=0 solve Schr. Eq Find average values P460 - dev. wave eqn.

16. P460 - dev. wave eqn.

17. Momentum vs. Position space • Can solve SE (find eigenvalues and functions, make linear series) in either position or momentum space • Fourier transforms allow you to go back and forth - pick whichever is easiest P460 - dev. wave eqn.

18. Momentum vs. Position spaceexample • Expectation value of momentum in momentum space • integrate by parts and flip integrals P460 - dev. wave eqn.

19. Probability Current • Define probability density and probability current. Good for V real • gives conservation of “probability” (think of a number of particles, charge). Probability can move to a different x • V imaginary gives P decreasing with time (absorption model) P460 - dev. wave eqn.

20. Probability and Current Definitions With V real Use S.E. to substitute for substitute into integral and evaluate The wave function must go to 0 at infinity and so this is equal 0 P460 - dev. wave eqn.

21. Probability Current Example • Supposition of 2 plane waves (right-going and left-going) P460 - dev. wave eqn.