Problem 1.17 A particle is represented (at time=0) by the wave function A(a²-x²). if-a ≤x≤+a. 0, otherwise. 4(x, 0) = [ (a) Determine the normalization constant A. (b) What is the expectation value of x (at time t = 0)? (c) What is the expectation value of p (at time t = 0)? (Note that you cannot get it from p = md(x)/dt. Why not?) (d) Find the expectation value of x². (e) Find the expectation value of p².
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- Problem 2.13 A particle in the harmonic oscillator potential starts out in the state ¥ (x. 0) = A[3¥o(x)+ 4¼1(x)]. (a) Find A. (b) Construct ¥ (x, t) and |¥(x. t)P. (c) Find (x) and (p). Don't get too excited if they oscillate at the classical frequency; what would it have been had I specified ¥2(x), instead of Vi(x)? Check that Ehrenfest's theorem (Equation 1.38) holds for this wave function. (d) If you measured the energy of this particle, what values might you get, and with what probabilities?Problem 2.21 Suppose a free particle, which is initially localized in the range -a < x < a, is released at time t = 0: А, if -a < х <а, otherwise, (x, 0) = where A and a are positive real constants. 50 Chap. 2 The Time-Independent Schrödinger Equation (a) Determine A, by normalizing V. (b) Determine (k) (Equation 2.86). (c) Comment on the behavior of (k) for very small and very large values of a. How does this relate to the uncertainty principle? *Problem 2.22 A free particle has the initial wave function (x, 0) = Ae ax where A and a are constants (a is real and positive). (a) Normalize (x, 0). (b) Find V(x, t). Hint: Integrals of the form e-(ax?+bx) dx can be handled by "completing the square." Let y = Ja[x+(b/2a)], and note that (ax? + bx) = y? – (b²/4a). Answer: 1/4 e-ax?/[1+(2ihat/m)] 2a Y (x, t) = VI+ (2iħat/m) (c) Find |4(x, t)2. Express your answer in terms of the quantity w Va/[1+ (2hat/m)²]. Sketch |V|? (as a function of x) at t = 0, and again for some very large t.…Problem 2.8 A particle of mass m in the infinite square well (of width a) starts out in the left half of the well, and is (at t = 0) equally likely to be found at any point in that region. (a) What is its initial wave function, (x, 0)? (Assume it is real. Don't forget to normalize it.) (b) What is the probability that a measurement of the energy would yield the value л²ħ²/2ma²?
- Suppose a particle has zero potential energy for x < 0. a constant value V. for 0 ≤ x ≤ L. and then zero for x > L. Sketch the potential. Now suppose that wavefunction is a sine wave on the left of the barrier. declines exponentially inside the barrier. and then becomes a sine wave on the right. beingcontinuous everywhere. Sketch the wavefunction on your sketch of the potential energy.Problem 2.34 Consider the "step" potential:53 V (x) = [0, x ≤0, Vo, x > 0. (a) Calculate the reflection coefficient, for the case E VoWhat is the answer of question 2
- By employing the prescribed definitions of the raising and lowering operators pertaining to the one-dimensional harmonic oscillator: x = ħ 2mω -(â+ + â_) hmw ê = i Compute the expectation values of the following quantities for the nth stationary staten. Keep in mind that the stationary states form an orthogonal set. 2 · (â+ − â_) [ pm 4ndx YmVndx = 8mn a. The position of particle (x) b. The momentum of the particle (p). c. (x²) d. (p²) e. Confirm that the uncertainty principle is satisfied for all values of n11. Evaluate (r), the expectation value of r for Y,s (assume that the operator f is defined as "multiply by coordinate r).Why does (r) not equal 0.529 for Y,,? In this problem,use 4ardr = dt.For the Osaillator problem, mwx2 har monit (부) (2M) y e Y, LX) = Mw 1. Use the lowering operator to find Yo(X). 2. Is your wave function normalized ? Check.
- 2.4. A particle moves in an infinite cubic potential well described by: V (x1, x2) = {00 12= if 0 ≤ x1, x2 a otherwise 1/2(+1) (a) Write down the exact energy and wave-function of the ground state. (2) (b) Write down the exact energy and wavefunction of the first excited states and specify their degeneracies. Now add the following perturbation to the infinite cubic well: H' = 18(x₁-x2) (c) Calculate the ground state energy to the first order correction. (5) (d) Calculate the energy of the first order correction to the first excited degenerated state. (3) (e) Calculate the energy of the first order correction to the second non-degenerate excited state. (3) (f) Use degenerate perturbation theory to determine the first-order correction to the two initially degenerate eigenvalues (energies). (3)The Hamiltonian of a one-dimensional harmonic oscillator can be written in natural units (m = hbar= ω = 1) as: (image1) Where ˆa =(ˆx+ipˆ)/√2, and ˆa† =(ˆx−ipˆ)/√2 One of the proper functions is: (image2) Find the two eigenfunctions closest in energy to the function ψa (you don't have to normalize) .1.3. Determine an orthogonal basis for the subspace of C (-1, 1] spanned by functions: {f(x) = x, f(x) = x³, f(x) = x³] using Gram-Schmidt process.