Solving Schrodinger Equation: ”Easy” Bound States If V(x,t)=v(x) than can separate variables G is separation constant valid any x or t. note h is really hbar Gives 2 ordinary diff. Eqns. 1

Solutions to Schrod Eqn Gives energy eigenvalues and eigenfunctions (wave functions). These are quantum states. Linear combinations of eigenfunctions are also solutions. For discrete solutions 2

G=E if 2 energy states, interference/oscillation 1D time independent Scrod. Eqn. Solve: know U(x) and boundary conditions want mathematically well-behaved. Do not want: No discontinuities. Usually except if V=0 or y =0 in certain regions 3

Solutions to Schrod Eqn Linear combinations of eigenfunctions are also solutions. Assume two energies assume know wave function at t=0 at later times the state can oscillate between the two states - probability to be at any x has a time dependence P460 - Sch. wave eqn. 4

Example 3-1 Boundary conditions (including the functions being mathematically well behaved) can cause only certain, discrete eigenfunctions solve eigenvalue equation impose the periodic condition to find the allowed eigenvalues P460 - Sch. wave eqn. 5

Boundary condition is that y is continuous:give: Square Well Potential Start with the simplest potential Boundary condition is that y is continuous:give: V Draw wave functions; readily get wavelength p=h/l -a/2 a/2 6

Infinite Square Well Potential Solve S.E. where V=0 Boundary condition quanitizes k/E, 2 classes Even y=Bcos(knx) kn=np/a n=1,3,5... y(x)=y(-x) Odd y=Asin(knx) kn=np/a n=2,4,6... y(x)=-y(-x) Parity operator Note:shift x-axis 7

Infinite Square Well Potential Need to normalize the wavefunction. Look up in integral tables What is the minimum energy of an electron confined to a nucleus? Let a = 10-14m = 10 F 8

Infinite Square Well Density of States The density of states is an important item in determining the probability that an interaction or decay will occur it is defined as for the infinite well For electron with a = 1mm, what is the number of states within 0.0001 eV about 0.01 eV? 9

Example Particle in box with width a and a wavefunction of Find the probability that a measurement of the energy gives the eigenvalue En With only n=odd only from the symmetry The probability to be in state n is then (essentially dot product) 10

Example Particle in box with width a and a wavefunction of psi(x) = sqrt(4/5)u_1(x) + sqrt(1/5)u_2(x) What is <E>? <E> = 4/5 E_1 + 1/5 E_2 = h^2/8ma^2(4/5 + 1/5*4) Draw the wavefunction. Is <x> non-zero? 11

Finite Square Well Potential For V=finite “outside” the well. Solutions to S.E. inside the well the same. Have different outside. The boundary conditions (wavefunction and its derivative continuous) give quantization for E<V 0 longer wavelength, lower Energy. Finite number of energy levels Outside:

Boundary Condition Want wavefunction and its derivative to be continuous often a symmetry such that solution at +a also gives one at -a Often can do the ratio (see book which solves) and that can simplify the algebra

Finite Square Well Potential:mostly skip Equate wave function at boundaries And derivative

Finite Square Well:mostly skip Harris does algebra. 2 classes. Solve numerically k1 and k2 both depend on E. Quantization sets allowed energy levels

Finite Square Well Potential:remember this Number of bound states is finite. Calculate assuming “infinite” well energies. Get n. Add 1 Electron V=100 eV width=0.2 nm Deuteron p-n bound state. Binding energy 2.2 MeV radius = 2.1 F (really need 3D S.E………)

Finite Square Well Potential:remember this Can do an approximation by guessing at the penetration distance into the “forbidden” region. Use to estimate wavelength Electron V=100 eV width=0.2 nm d

Delta Function Potential:skip d-function can be used to describe potential. Assume attractive potential V and E<V bound state. Potential has strength l/a. Rewrite Schro.Eq V

Delta Function Potential II:skip Except for x=0 have exponential solutions. Continuity condition at x=0 is (in some sense) on the derivative. See by integrating S.E in small region about x=0 V

Harmonic Oscillators V F=-kx or V=cx2. Arises often as first approximation for the minimum of a potential well Solve directly through “calculus” (analytical). Do in 460 Solve using group-theory like methods from relationship between x and p (algebraic). Do in 460 V Classically F=ma and Sch.Eq.

Harmonic Oscillators-Guess Can use our solution to finite well to guess at a solution Know lowest energy is 1-node, second is 2-node, etc. Know will be Parity eigenstates (odd, even functions) Could try to match at boundary but turns out in this case can solve the diff.eq. for all x - as no abrupt changes in V(x) V y

Solve Hermite Equation Get recursion relationship. Skip if any al=0 then the series can end (higher terms are 0). Gives eigenvalues easiest if define odd or even types. Need to know energy eigenvalues, but not how to derive

Hermite Equation - wavefunctions Use recursion relationship to form eigenfunctions. Compare to first 2 for infinite well First Second Third