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- 4. For parts a, b, c state whether the given wavefunction is admissible or not and justify your answer. (a) Given y (x, t) = Cei(kx-wt) + Cze-i(kx+wt) with C1 and C2 complex numbers and k real. (b) Given 4 (x, t) = A cos(@t) e-blal with A and b real, and b > 0. (c) Given (x, t) = Ce-iwte-alx| for -o 0. (d) Sketch the probability density associated with the wavefunction p(x, t) defined in part (c).8. If the wave function has the following equation y = 4cosA + 4isinA a. Determine the conjugate b. |y*y|1- Considera particle whose normalized wavefunction is given by (x) 2a/2xe x for x > 0 and (x) = 0 for x < 0. For what values does the l(x)[? have a peak? b- Calculate (x) and (x) %3D a-
- 2. Suppose a particle of mass, m, has energy E, and wave function: WE (x, t = 0) = Aeikx + Be-ikx What is WE(x, t)? Calculate the probability density of the particle when it has the wavefunction, E (x, t). If you wish to simply your answer algebraically, use this information: let A = aeia and B = beif and A = a - B. where the variables a, b, a, ß are all real! WARNING: Simplifying the equation is somewhat time consuming. Show/explain why p(x,t) is not normalizable. According to quantum theory, all physical systems must have an associated wavefunction that is normalizable. Explain why plane wave solutions to Schrodinger's equation are used in quantum theory, despite not being normalizable?4) Energy measurements A wavefunction of a particle is written as the superposition of stationary states: 1 Y(x,t = 0) = ; (u, (x) + (1 – i)u2(x) + c3U3(x)) | with corresponding energies E, = 1 eV, E2 = 2 eV, and Ez = 3 eV. %3D a) Determine c3. Is it unique? Briefly explain your answer. b) Determine the expectation value of the total energy, (E), and explain how it will change with time. c) What is the probability that a single measurement of the energy in this particular case will result in the expectation value of the energy you found in (b)? Briefly explain your answer.16. Consider the wave function Mx) = (alT)" exp(-ax12) Calculate (x) for n = 1, 2. Can you quickly write down the result for (")?
- 2. The for dimensional oscillator is wave one w(x) = Ae-a²x² /2, for the ground state with energy %3D Eo ho where a? when a perturbation 2' V(t) = E, is applied to the first order change 10 %3D in the ground state energy is Eo (a) x 10-4 (b) 3E, x 10-4 3 (c) E, x 10-4 (d) E,×10-415. For the wavefunction (r) = Nre 22, where is the most probable location of the particle? 16. Write the time-independent Schrodinger equation for a particle sub- jected to a constant uniform force.Q4. Proof that the probability current for a wavefunction of the form Þ(x) = R(x) eis(x) ħ əs(x) (a) j = |R(x)|²; т дх (b) j = ħk |R|2 for a wavevector k. Q5. Proof the following: (a) Using the Cartesian components of L, Proof that: [L,. Lz]y = 0 (b) Lž + L? # (Ly + iL,)(Ly – iLz) (c) [L². L+] = 0
- 6. An electron in hydrogen atom is in initial state Þ(r, 0) = A(2410o + iÞ210 + 421–1 – 2ib211) where wnim are the eigenfunctions of the hydrogen atom a. Determine the constant A b. What is the probability of finding the electron in the first excited state? hw = - n2 c. Write the state Þ(r, t) at time t, using energy eigenvalues as En d. Find the expectation value of L in the state Þ(r,t e. Find the expectation values of Lx and Ly in the state (r, t f. If measurement of Lz led to the value –ħ what will be results of measurement of energy and the square of total orbital momentum immediately afterwards and what are their probabilities?The wavefunction of a 1D PIB is given by: 4₁-415567 sin([π/a] x) what is the box length in pm? O a. 7.82 O b. 27.2 O c. 16.8 O d. 19.7 O e. 11.65:30 PM Mon Sep 26 11 тод 3. Normalize the following 2s wavefunction: (r, 0, Ф) = (2 where 0SEE MORE QUESTIONS