1. A particle of mass m is moving in one dimension in a potential
V(x, t). The wave function for the particle isfor −∞ x k and A are
constants.(a) Show that V is independent of t, and determine V(x).(b)
Normalize this wave function.(c) Using the normalized wave function,
calculate 〈x〉, 〈x2〉, 〈p〉, and 〈p2〉. Get solution
2. Determine which of the following one-dimensional wave functions represent states of definite momentum. For each wave function that does correspond to a state of definite momentum, determine the momentum.(a) ψ(x) = eikx(b) ψ(x) = xe'ikx(c) ψ(x) = sin(kx) + i cos(kx)(d) ψ(x) = eikx + e−ikx Get solution
3. The wave function for a particle is Ψ(x, t) = sin(kx)[i cos(ωt/2) + sin(ωt/2)], where k and ω are constants.(a) Is this particle in a state of definite momentum? If so, determine the momentum.(b) Is this particle in a state of definite energy? If so, determine the energy. Get solution
4. A particle with mass m is, moving in one dimension near the speed of light so that the relation...for the kinetic energy is no longer valid. Instead, the total energy is given by...Hence, we can no longer use the Schrödinger equation. Suppose the wave function Ψ(x, t) for the particle is an eigenfunction of the energy operator and an eigenfunction of the momentum operator, and also assume that there is no potential energy V. Derive a linear differential equation for Ψ(x, t). Get solution
5. A particle with mass m is moving in one dimension in the potential V(x). The particle is in a state of definite energy E, but it is not in a state of definite momentum p. Show that... Get solution
6. Consider the solution to the Schrödinger equation for the infinite square well with n = 2 rather than n = 1 in Equation (3.20). Derive Ψ(x, t) for this case, and normalize this wave function. Get solution
7. Suppose that a wave function Ψ (r, t) is normalized. Show that the wave function eiθΨ(r, t), where θ is an arbitrary real number, is also normalized. Get solution
8. Suppose that ψ1 and ψ2 are two different solutions of the time-independent Schrödinger equation with the same energy E.(a) Show that ψ1 + ψ2 is also a solution with energy E.(b) Show that cψ1 is also a solution of the Schrödinger equation with energy E. Get solution
9. A particle moves in one dimension in the potential shown here. The energy E is shown on the graph, and the particle is in its ground state....(a) Sketch ψ(x) for this particle.(b) You make a measurement to find the particle. Indicate on your graph the point or points at which you are most likely to find it. Get solution
10. A particle moves in one dimension in the potential shown here. The energy E is shown on the graph, and the particle is in its first excited state....(a) Sketch ψ(x) for this particle.(b) You make a measurement to find the particle. Indicate on your graph the point or points at which you are most likely to find it. Get solution
11. A particle moving in one dimension is described by the function ψ(x) shown here:...(a) You make a measurement to locate the particle. Which one of the following is true?i. You will always find the particle at point B.ii. You are most likely to find the particle at points A or C and least likely to find the particle at point B.iii. You are most likely to find the particle at points A, B, or C.Explain your answer.(b) Which one of the following potentials V(x) could give rise to this ψ(x)?... Get solution
2. Determine which of the following one-dimensional wave functions represent states of definite momentum. For each wave function that does correspond to a state of definite momentum, determine the momentum.(a) ψ(x) = eikx(b) ψ(x) = xe'ikx(c) ψ(x) = sin(kx) + i cos(kx)(d) ψ(x) = eikx + e−ikx Get solution
3. The wave function for a particle is Ψ(x, t) = sin(kx)[i cos(ωt/2) + sin(ωt/2)], where k and ω are constants.(a) Is this particle in a state of definite momentum? If so, determine the momentum.(b) Is this particle in a state of definite energy? If so, determine the energy. Get solution
4. A particle with mass m is, moving in one dimension near the speed of light so that the relation...for the kinetic energy is no longer valid. Instead, the total energy is given by...Hence, we can no longer use the Schrödinger equation. Suppose the wave function Ψ(x, t) for the particle is an eigenfunction of the energy operator and an eigenfunction of the momentum operator, and also assume that there is no potential energy V. Derive a linear differential equation for Ψ(x, t). Get solution
5. A particle with mass m is moving in one dimension in the potential V(x). The particle is in a state of definite energy E, but it is not in a state of definite momentum p. Show that... Get solution
6. Consider the solution to the Schrödinger equation for the infinite square well with n = 2 rather than n = 1 in Equation (3.20). Derive Ψ(x, t) for this case, and normalize this wave function. Get solution
7. Suppose that a wave function Ψ (r, t) is normalized. Show that the wave function eiθΨ(r, t), where θ is an arbitrary real number, is also normalized. Get solution
8. Suppose that ψ1 and ψ2 are two different solutions of the time-independent Schrödinger equation with the same energy E.(a) Show that ψ1 + ψ2 is also a solution with energy E.(b) Show that cψ1 is also a solution of the Schrödinger equation with energy E. Get solution
9. A particle moves in one dimension in the potential shown here. The energy E is shown on the graph, and the particle is in its ground state....(a) Sketch ψ(x) for this particle.(b) You make a measurement to find the particle. Indicate on your graph the point or points at which you are most likely to find it. Get solution
10. A particle moves in one dimension in the potential shown here. The energy E is shown on the graph, and the particle is in its first excited state....(a) Sketch ψ(x) for this particle.(b) You make a measurement to find the particle. Indicate on your graph the point or points at which you are most likely to find it. Get solution
11. A particle moving in one dimension is described by the function ψ(x) shown here:...(a) You make a measurement to locate the particle. Which one of the following is true?i. You will always find the particle at point B.ii. You are most likely to find the particle at points A or C and least likely to find the particle at point B.iii. You are most likely to find the particle at points A, B, or C.Explain your answer.(b) Which one of the following potentials V(x) could give rise to this ψ(x)?... Get solution
