Find the expectation value of the energy of the state defined by Y = Po, where , is an eigenstate of the one-dimensional harmonic oscillator (arbitrary v) and P is the reduced momentum operator, P = Applying the harmonic oscillator raising and lowering operators

Estimate the lowest state energy of a system with Hamiltonian, H =- +V(x) where 2m V(x) = 9£x for r>0 0 for xs0 -bx Choose the Fang-Howard wavefunction w(x) = x e 2 as trial wave function.

Consider the potential barrier problem as illustrated in the figure below. Considering the case where E > V0: (a) find the wave function up to a constant (that is, you don't need to compute the normalization constant) (b) Calculate the reflection coefficient of the wave function. This result is expected classically?

Consider the scattering of a particle by a regular lattice of basis a, b, c. The interaction with the lattice can be written as V = E, V(Ir – r,|). where V (r-r,|) is the potential of each atom and is spherically symmetric about the atom's lattice point. Show using the Born approximation that the condition for non-vanishing scattering is that the Bragg law be satisfied.

Show that the following wave function is normalized. Remember to square it first. Limits of integration go from -infinity to infinity. DO NOT SKIP ANY STEPS IN THE PROCEDURE

Consider a quantum state J- √64-1 + √241 +242 that is a superposition of three eigenstates of operator with eigenvalues w_1 = −1, W1 = 1, and w2 = 2 (same as subscripts k in above). The expectation value of is OO V 6 1 √3+2√2-1 √6

Write a matrix representation for position and momentum operators on bases made of eigenstates oscillator modes.

Let ynlm denote the eigenfunctions of a Hamiltonian for a spherically symmetric potential M7). The expectation value of L, in the state w+5 210 + v10 y-1 + /20 y 21, is %D

Suppose that the wave function for a system can be written as 4(x) = √3 4 · Φι(x) + V3 2√₂ $2(x) + 2 + √3i 4 $3(x) and that 1(x), 2(x), and 3(x) are orthonormal eigenfunc- tions of the operator Ekinetic with eigenvalues E₁, 2E₁, and 4E₁, respectively. a. Verify that (x) is normalized. b. What are the possible values that you could obtain in measuring the kinetic energy on identically prepared systems? c. What is the probability of measuring each of these eigenvalues? d. What is the average value of Ekinetic that you would obtain from a large number of measurements?

A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: (x,0) = A [v₁ (x) + ₂(x)]. (a) Normalize (x, 0). (That is, find A. This is very easy, if you exploit the orthonormality of 1 and 2. Recall that, having normalized at t = 0, you can rest assured that it stays normalized—if you doubt this, check it explicitly after doing part (b).) (c) (b) Find (x, t) and (x, t)|². Express the latter as a sinusoidal function of time, as in Example 2.1. To simplify the result, let w = ²ħ/2ma². Compute (x). Notice that it oscillates in time. What is the angular frequency of the oscillation? What is the amplitude of the oscillation? (If your amplitude is greater than a /2, go directly to jail.) (d) Compute (p). (As Peter Lorre would say, "Do it ze kveek vay, Johnny!”) (e) If you measured the energy of this particle, what values might you get, and what is the probability of getting each of them? Find the expectation value of H. How does…