1. Show that the differential equation solution given in Equation
(4.5), ......, is completely equivalent to the solution in Equation
(4.6), ......, and express C1 and C2 in terms of D1 and D2. If both D1
and D2 are real, is it possible for both C1 and C2 to be real? Get solution
2. Show that the general expression for the wave function for a free particle, given by Equation (4.8) as ..., is not an eigenfunction of momentum unless C1 = 0 or C2 = 0. Get solution
3. A particle with mass m and energy E is moving in one dimension from right to left. It is incident on the step potential V(x) = 0 for x V(x) = V0 for x ≥ 0, where V0 > 0, as shown on the diagram. The energy of the particle is E > V0....(a) Solve the Schrödinger equation to derive the wave function for x x ≥ 0. Express the solution in terms of a single unknown constant.(b) Calculate the value of the reflection coefficient R for the particle. Get solution
4. A particle with mass m and energy E is moving in one dimension from left to right. It is incident on the step potential V(x) = 0 for x V(x) = V0 for x ≥ 0, where V0 > 0, as shown on the diagram. The energy of the particle is exactly equal to V0, i.e., E = V0....(a) Solve the Schrödinger equation to derive the wave function for x x ≥ 0. Express the solution in terms of a single unknown constant.(b) Calculate the value of the reflection coefficient R for the particle. Get solution
5. Consider reflection from a step potential of height V0 with E > V0, but now with an infinitely high wall added at a distance a from the step (see diagram):...(a) Solve the Schrödinger equation to find ψ(x) for x x ≤ a. Your solution should contain only one unknown constant.(b) Show that the reflection coefficient at x = 0 is R = 1. This is different from the value of R previously derived without the infinite wall. What is the physical reason that R = 1 in this case?(c) Which part of the wave function represents a leftward-moving particle at x ≤ 0? Show that this part of the wave function is an eigenfunction of the momentum operator, and calculate the eigenvalue. Is the total wave function for x ≤ 0 an eigenfunction of the momentum operator? Get solution
6. An electron is accelerated through a potential difference of 3 eV and is incident on a finite potential barrier of height 5 eV and thickness 5 × 10−10 m. What is the probability that the electron will tunnel through the barrier? Get solution
7. Consider an infinite square-well potential of width a, but with the coordinate system shifted so that the infinite potential barriers lie at x = −a/2 and x = a/2 (see diagram):...(a) Solve the Schrödinger equation for this case to calculate the normalized wave functions ψn(x) and the corresponding energies En.(b) Explain why you get the same energies as for the square well between x = 0 and x = a, but a different set of wave functions. Get solution
8. A baseball (see Example 4.2) is confined between two thick walls a distance 0.5 m apart. Calculate the zero-point energy of the baseball. Get solution
9. A particle is trapped inside an infinite one-dimensional square well of width a in the first excited state (n = 2).(a) You make a measurement to locate the particle. At what positions are you most likely to find the particle? At what positions are you least likely to find it?(b) Calculate 〈p2〉 for this particle. Get solution
10. A particle is bound in a one-dimensional potential V(x), where V(x) is symmetric, i.e., V(x) = V(−x).(a) Suppose that Ψ(x) is a solution of the Schrödinger equation with energy E. Make the change of variables y = −x, and show that ψ(y) is also a solution of the Schrödinger equation with energy E.(b) Since the solutions of the Schrödinger equation for a fixed value of E are unique (up to multiplication by a constant), the result from part (a) implies that ψ(x) = cψ(−x), where c is an unknown constant. Use this result to show that ψ(x) must be either even [ψ(−x) = ψ(x)] or odd [ψ(−x) = −ψ(x)].(c) For a particle bound in a one-dimensional symmetric potential, so that V(−x) = V(x), show that all of the following are true:i. ψ*ψ is a symmetric function,ii. 〈x〉 = 0,iii. 〈p〉 = 0. Get solution
11. Consider the semi-infinite square well given by V(x) = −V0 x ≤ a and V(x) = 0 for x > a. There is an infinite barrier at x = 0 (hence the name “semi-infinite”). A particle with mass m is in a bound state in this potential with energy E ≤ 0.(a) Solve the Schrödinger equation to derive ψ(x) for x ≥ 0. Use the appropriate boundary conditions and normalize the wave function so that the final answer does not contain any arbitrary constants.(b) Show that the allowed energy levels E must satisfy the equation...(c) The equation in part (b) cannot be solved analytically to give the allowed energy levels, but simple solutions exist in certain special cases. Determine the conditions on V0 and a so that a bound state exists with E = 0. Get solution
12. A particle of mass m moves in a harmonic oscillator potential. The particle is in the first excited state.(a) Calculate 〈x〉 for this particle.(b) Calculate 〈p〉 for this particle.(c) Calculate 〈p2〉 for this particle.(d) At what positions are you most likely to find the particle? At what positions are you least likely to find it? Get solution
13. The oscillation frequencies of a diatomic molecule are typically 1012 Hz−1014 Hz. Derive an order of magnitude estimate for the harmonic oscillator constant K for such molecules. Get solution
14. A particle of mass m is bound in a one-dimensional power law potential V(x) = Kxβ, where ß is an even positive integer. Show that the allowed energy levels are proportional to m−β/(2+β). Get solution
15. A particle is moving in a simple harmonic oscillator potential ... for x ≥ 0, but with an infinite potential barrier at x = 0 (the paddle ball potential). Calculate the allowed wave functions and corresponding energies. Get solution
16. A particle moves in one dimension in the potential V(x) = V0 ln(x/x0) for x > 0, where x0 and V0 are constants with units of length and energy, respectively. There is an infinite potential barrier at x = 0. The particle drops from the first excited state with energy E1 into the ground state with energy E0, by emitting a photon with energy E1−E0. Show that the frequency of the photon emitted by this particle is independent of the mass of the particle. Get solution
2. Show that the general expression for the wave function for a free particle, given by Equation (4.8) as ..., is not an eigenfunction of momentum unless C1 = 0 or C2 = 0. Get solution
3. A particle with mass m and energy E is moving in one dimension from right to left. It is incident on the step potential V(x) = 0 for x V(x) = V0 for x ≥ 0, where V0 > 0, as shown on the diagram. The energy of the particle is E > V0....(a) Solve the Schrödinger equation to derive the wave function for x x ≥ 0. Express the solution in terms of a single unknown constant.(b) Calculate the value of the reflection coefficient R for the particle. Get solution
4. A particle with mass m and energy E is moving in one dimension from left to right. It is incident on the step potential V(x) = 0 for x V(x) = V0 for x ≥ 0, where V0 > 0, as shown on the diagram. The energy of the particle is exactly equal to V0, i.e., E = V0....(a) Solve the Schrödinger equation to derive the wave function for x x ≥ 0. Express the solution in terms of a single unknown constant.(b) Calculate the value of the reflection coefficient R for the particle. Get solution
5. Consider reflection from a step potential of height V0 with E > V0, but now with an infinitely high wall added at a distance a from the step (see diagram):...(a) Solve the Schrödinger equation to find ψ(x) for x x ≤ a. Your solution should contain only one unknown constant.(b) Show that the reflection coefficient at x = 0 is R = 1. This is different from the value of R previously derived without the infinite wall. What is the physical reason that R = 1 in this case?(c) Which part of the wave function represents a leftward-moving particle at x ≤ 0? Show that this part of the wave function is an eigenfunction of the momentum operator, and calculate the eigenvalue. Is the total wave function for x ≤ 0 an eigenfunction of the momentum operator? Get solution
6. An electron is accelerated through a potential difference of 3 eV and is incident on a finite potential barrier of height 5 eV and thickness 5 × 10−10 m. What is the probability that the electron will tunnel through the barrier? Get solution
7. Consider an infinite square-well potential of width a, but with the coordinate system shifted so that the infinite potential barriers lie at x = −a/2 and x = a/2 (see diagram):...(a) Solve the Schrödinger equation for this case to calculate the normalized wave functions ψn(x) and the corresponding energies En.(b) Explain why you get the same energies as for the square well between x = 0 and x = a, but a different set of wave functions. Get solution
8. A baseball (see Example 4.2) is confined between two thick walls a distance 0.5 m apart. Calculate the zero-point energy of the baseball. Get solution
9. A particle is trapped inside an infinite one-dimensional square well of width a in the first excited state (n = 2).(a) You make a measurement to locate the particle. At what positions are you most likely to find the particle? At what positions are you least likely to find it?(b) Calculate 〈p2〉 for this particle. Get solution
10. A particle is bound in a one-dimensional potential V(x), where V(x) is symmetric, i.e., V(x) = V(−x).(a) Suppose that Ψ(x) is a solution of the Schrödinger equation with energy E. Make the change of variables y = −x, and show that ψ(y) is also a solution of the Schrödinger equation with energy E.(b) Since the solutions of the Schrödinger equation for a fixed value of E are unique (up to multiplication by a constant), the result from part (a) implies that ψ(x) = cψ(−x), where c is an unknown constant. Use this result to show that ψ(x) must be either even [ψ(−x) = ψ(x)] or odd [ψ(−x) = −ψ(x)].(c) For a particle bound in a one-dimensional symmetric potential, so that V(−x) = V(x), show that all of the following are true:i. ψ*ψ is a symmetric function,ii. 〈x〉 = 0,iii. 〈p〉 = 0. Get solution
11. Consider the semi-infinite square well given by V(x) = −V0 x ≤ a and V(x) = 0 for x > a. There is an infinite barrier at x = 0 (hence the name “semi-infinite”). A particle with mass m is in a bound state in this potential with energy E ≤ 0.(a) Solve the Schrödinger equation to derive ψ(x) for x ≥ 0. Use the appropriate boundary conditions and normalize the wave function so that the final answer does not contain any arbitrary constants.(b) Show that the allowed energy levels E must satisfy the equation...(c) The equation in part (b) cannot be solved analytically to give the allowed energy levels, but simple solutions exist in certain special cases. Determine the conditions on V0 and a so that a bound state exists with E = 0. Get solution
12. A particle of mass m moves in a harmonic oscillator potential. The particle is in the first excited state.(a) Calculate 〈x〉 for this particle.(b) Calculate 〈p〉 for this particle.(c) Calculate 〈p2〉 for this particle.(d) At what positions are you most likely to find the particle? At what positions are you least likely to find it? Get solution
13. The oscillation frequencies of a diatomic molecule are typically 1012 Hz−1014 Hz. Derive an order of magnitude estimate for the harmonic oscillator constant K for such molecules. Get solution
14. A particle of mass m is bound in a one-dimensional power law potential V(x) = Kxβ, where ß is an even positive integer. Show that the allowed energy levels are proportional to m−β/(2+β). Get solution
15. A particle is moving in a simple harmonic oscillator potential ... for x ≥ 0, but with an infinite potential barrier at x = 0 (the paddle ball potential). Calculate the allowed wave functions and corresponding energies. Get solution
16. A particle moves in one dimension in the potential V(x) = V0 ln(x/x0) for x > 0, where x0 and V0 are constants with units of length and energy, respectively. There is an infinite potential barrier at x = 0. The particle drops from the first excited state with energy E1 into the ground state with energy E0, by emitting a photon with energy E1−E0. Show that the frequency of the photon emitted by this particle is independent of the mass of the particle. Get solution
