(a)
The expectation value of
(a)
Answer to Problem 3.42P
The expectation value of
Explanation of Solution
Write the expression for the expectation value of the position.
Here,
Write the expression for the
Write the expression for the expectation value of momentum.
Write the expectation value of
Conclusion:
Therefore, the expectation value of
(b)
The value of
(b)
Answer to Problem 3.42P
The value of
Explanation of Solution
Write the expression for the
Use equation (I) and (II) to solve for
Write the expression for
Use equation (III) and (IV) to solve for
Use equation (VII) and (VIII) to find
Conclusion:
Therefore, the value of
(c)
Show that the expansion coefficients are
(c)
Answer to Problem 3.42P
It is showed that the expansion coefficients are
Explanation of Solution
Write the expression for the
Conclusion:
Therefore, it is showed that the expansion coefficients are
(d)
The value of
(d)
Answer to Problem 3.42P
The value of
Explanation of Solution
Write the expression for the normalization of
Conclusion:
Therefore, the value of
(e)
Show that
(e)
Answer to Problem 3.42P
It is showed that
Explanation of Solution
Write the expression for
Apart from the overall phase factor
Conclusion:
Therefore, it is showed that
(f)
The value of
(f)
Answer to Problem 3.42P
The value of
Explanation of Solution
It is given that the value of
Use equation (I) to solve for the value of
Conclusion:
Therefore, the value of
(g)
Whether the ground state
(g)
Answer to Problem 3.42P
Yes, the ground state
Explanation of Solution
Write the expression for the given value of
From equation (XIV), it is known that the ground state
Conclusion:
Therefore, the ground state
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Chapter 3 Solutions
Introduction To Quantum Mechanics
- conditions.) Problem 2.4 Solve the time-independent Schrödinger equation with appropriate boundary conditions for an infinite square well centered at the origin [V (x) = 0, for -a/2 < x < +a/2; V (x) = ∞ otherwise]. Check that your allowed energies are consistent with mine (Equation 2.23), and confirm that your y's can be obtained from mine (Equation 2.24) by the substitution x x - a/2. Droblo m 25 Celaulnte lu) .2arrow_forwardDivergence theorem. (a) Use the divergence theorem to prove, v = -478 (7) (2.1) (b) [Problem 1.64, Griffiths] In case you're not persuaded with (a), try replacing r by (r² + e²)2 and watch what happens when ɛ → 0. Specifically, let 1 -V². 4л 1 D(r, ɛ) (2.2) p2 + g2 By taking note of the defining conditions of 8°(7) [(1) at r = 0, its value goes to infinity, (2) for all r + 0, its value is 0, and (3) the integral over all space is 1], demonstrate that 2.2 goes to 8*(F) as ɛ → 0.arrow_forwardDetermine the transmission coefficient for a rectangular barrier (same as Equation 2.127, only with +Vo in the region -a Vo (note that the wave function inside the barrier is different in the three cases). Partial answer: For Earrow_forwardProblem 3.10 Is the ground state of the infinite square well an eigenfunction of momentum? If so, what is its momentum? If not, why not?arrow_forwardA particle of mass in moving in one dimension is confined to the region 0 < 1 < L by an infinite square well potential. In addition, the particle experiences a delta function potential of strengtlh A located at the center of the well (Fig. 1.11). The Schrödinger equation which describes this system is, within the well, + A8 (x – L/2) v (x) == Ep(x), 0 < x < L. !! 2m VIx) L/2 Fig. 1.11 Find a transcendental equation for the energy eigenvalues E in terms of the mass m, the potential strength A, and the size L of the system.arrow_forwardWrite down the equations and the associated boundary conditions for solving particle in a 1-D box of dimension L with a finite potential well, i.e., the potential energy U is zero inside the box, but finite outside the box. Specifically, U = U₁ for x L. Assuming that particle's energy E is less than U, what form do the solutions take? Without solving the problem (feel free to give it a try though), qualitatively compare with the case with infinitely hard walls by sketching the differences in wave functions and probability densities and describing the changes in particle momenta and energy levels (e.g., increasing or decreasing and why), for a given quantum number.arrow_forwardThe 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) .arrow_forwardProblem 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?arrow_forwardProblem 3.27 Sequential measurements. An operator Ä, representing observ- able A, has two normalized eigenstates 1 and 2, with eigenvalues a1 and a2, respectively. Operator B, representing observable B, has two normalized eigen- states ø1 and ø2, with eigenvalues b1 and b2. The eigenstates are related by = (3ø1 + 402)/5, 42 = (401 – 302)/5. (a) Observable A is measured, and the value aj is obtained. What is the state of the system (immediately) after this measurement? (b) If B is now measured, what are the possible results, and what are their probabilities? (c) Right after the measurement of B, A is measured again. What is the proba- bility of getting a¡? (Note that the answer would be quite different if I had told you the outcome of the B measurement.)arrow_forwardProblem 2.7 A particle in the infinite square well has the initial wave function JAx, У (х, 0) — 0< x < a/2, a/2 < x < a. А (а — х), (a) Sketch ¥ (x, 0), and determine the constant A. (b) Find ¥(x, t). (c) What is the probability that a measurement of the energy would yield the value E1? (d) Find the expectation value of the energy, using Equation 2.21.21arrow_forward2.4. 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)arrow_forwardProblem #1 (Problem 5.3 in book). Come up with a function for A (the Helmholtz free energy) and derive the differential form that reveals A as a potential: dA < -SdT – pdV [Eqn 5.20]arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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