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.

vel=<x(t2)>-<x(t1)> 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.

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 momentum space are conjugate the spatial function implies “something” about the function in momentum space P460 - dev. wave eqn.

Wave Functions (time) If a wave is moving in the x-direction (or -x) with wave number k 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.

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 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.

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.

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.

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.

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.

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.

Continuous function look at “matrix” elements 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.

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.

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) P460 - dev. wave eqn.

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

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

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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.

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

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.

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.

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