Physics 443 Prelim # with solutions March 7, 8 Each problem is worth 34 points.. Harmonic Oscillator Consider the Hamiltonian for a simple harmonic oscillator H p m + mω x (a Use dimensional analysis to estimate the ground state energy and the characteristic size of the ground state wave function in terms of m,,and ω. (That is, determine the characteristic length l and energy E. [Let x l z and E ɛe, where l and E have dimensions of length and energy respectively. equation gives ( ml Substitution into Schrodinger s d dz + mω lz ψ ɛe ψ Multiply through by ml/ to get ( d dz + m ω l4 z ψ ɛ ml E ψ Define l so that the coefficient of z ψ on the left hand side is and E so that the coefficient of ɛψ on the right hand side is. Then l mω and E ω.] (b At t a particle in the harmonic oscillator potential has as its wave function an even mixture of the first two stationary states 3 with energies ω and ω. Ψ(x, A[ψ (x + ψ (x] Compute A and find Ψ(x, t and Ψ(x, t. Give your answers in terms of ψ and ψ. [A is the normilzation constant. ψ ψ A [ ψ ψ + ψ ψ + ψ ψ + ψ ψ A [ + + + ] A
Ψ(x, t [ψ (xe ie /t +ψ (xe ie /t ] [ψ (xe iωt/ +ψ (xe 3iωt/ ] Ψ(x, t [ + + ψ ψ e iωt + ψ ψ e ωt ] [ + ψ ψ cos(ωt] (c Compute x using x (a mω + + a. frequency of the oscillation? What is the angular [ x ψ x ψ ( ψ x ψ + ψ x ψ + ψ x ψ e iωt + ψ x ψ e iωt ( ψ x ψ e iωt + ψ x ψ e iωt ( ψ x ψ cos(ωt The last step follows from the fact that the wave functions ψ and ψ are real and x is an Hermitian operator. Finally ψ x ψ mω ψ a + + a ψ mω ψ a + ψ mω The last step follows from the third equation for the one dimensional harmonic oscillator on the formula sheet. So x mω cos(ωt. The angular frequency of the oscillation is ω.] (d If you measured the energy of this particle, what values might you get and what is the probability of getting each of them? [A measurement of the energy of the particle would give either 3 ω or ω with equal likelihood.]
(e What is the expectation value of H? [ H [ ψ H ψ + ψ H ψ ] ω]. WKB Consider the infinite square well with a sloped floor V (x kx for (x <, for ( < x < a,, for (x > a (a If the well is narrow (small a and k is small, the turning points for the ground state will be at x and x a. If the well is very broad, the right turning point for the ground state will occur along the floor, at x < a. Assuming that the turning points are at x and x < a, use the WKB approximation to find the energy of the ground state and the turning point. [The turning point x t E/k. The quantization condition for a well with one infinite vertical wall is (n 4 π x pdx x xt mk m(e kxdx xt ( E k x dx E mk( 3 k x3/ xt E mk( 3 k 3/ ( /3 3 3 πk E ] 4 m (b Now assume that the turning points are at x and x a and use the WKB approximation to write an expression that determines the energy levels of the system. (You do not need to solve for E n. [Now we use the quantization condition for two infinite 3
vertical walls. π x x a pdx mk m(e kxdx xt ( E k x dx E mk( 3 k x3/ a π ( (E 3/ ( E 3/ mk 3 k k a ] 3. Spin / Suppose that a spin-/ particle is in the state χ ( a b. where a cos α and b sin α are real and the state is normalized. (a What are the probabilities of getting +/ and /, if you measure S z and S x? [The probability for getting +/ if you measure along the z direction is ( χ a +χ ( a b and the probability for / is b. The probability for getting +/ if you measure along the x direction is χ (x + χ ( a ( b a + b. χ (x ± are the eigenvectors of S x with eigenvalues ±. The probability for / is a b.] (b In a coordinate system rotated by an angle θ about the y-axis so that z z and x x, what are the the probabilities of getting +/ and /, if you measure S z and S x? 4
[In the rotated coordinate system ( ( ( cos(θ/ sin(θ/ a a χ sin(θ/ cos(θ/ b b ( a cos(θ/ b sin(θ/ b cos(θ/ + a sin(θ/ Now the probability of getting +/ if you measure along the +z direction is a and the probability of getting / is b. The probability of getting ±/ if you measure along the +x axis is a ± b. 5
Formulae and Tables WKB x x p(xdx x x p(xdx x HΨ i Ψ t m x Ψ(x, t i Ψ(x, t t ( n π ( n π 4 (no infinite vertical walls ( infinite vertical wall x p(xdx nπ ( infinite vertical walls x p(x dx ψ(x A exp( ī x p(x dx + B exp( ī p h p h ψ(x C x exp( p(x p dx + D x exp( p(x p dx One dimensional harmonic oscillator H ˆp m + mωˆx a ± ( iˆp + mωˆx mω a + ψ n (n + ψ n+ a ψ n nψ n [a, a + ] H ω(a a + Hψ n ω(n + ψ n ψ ( mω /4 e mω x, ψ π ( mω π /4 mω mω xe x 6
Relativisitic energy momentum E p c + (mc Time dependence of an expectation value d Q dt ī [H, Q] + Q h t Three dimensional infinite cubical well Spin / ( 3/ ( ( ( nx π ψ nx,ny,n z (x, y, z sin a a x ny π sin a y nz π sin a z S x σ x E n x,n y,n z π (, S y σ y ma (n x + n y + n z ( i, S i z σ z (, (z R y (θ, (z, (z R y (θ, (z, (z R y (θ, (z, (z R y (θ, (z ( cos θ sin θ sin θ cos θ Virial Theorem For stationary states T r V Generators e i(σ ˆnφ/ cos(φ/ + i(ˆn σ sin(φ/ Boundary conditions for V (x αδ(x ψ(x continuous, ( dψ dx mαψ( 7