1. Suppose that the trial wave function ∣ψ(α)〉 happens to be exactly
equal to the true ground-state wave function ∣ψ0〉 for some value of α.
Show that in this case, the estimate of the ground-state energy given by
the variational principle will be equal to the true ground-state
energy. Get solution
2. Suppose that the trial wave function ∣ψ〉 used in the variational principle is orthogonal to the ground-state wave function of the Hamiltonian: 〈ψ0∣ψ(α)〉 = 0 for all values of α. Show that in this case...where E1 is the energy of the first excited state of H. Get solution
3. (a) In order to use the variational principle to estimate the ground-state energy of the one-dimensional potential V(x) = Kx4, where K is a constant, which of the following wave functions would be a better trial wave function?i. ...ii. ...Explain.(b) In order to use the variational principle to estimate the ground-state energy of the one-dimensional potential V(x) = Kx3 for x > 0 with an infinite potential barrier at x = 0, which of the following wave functions would be a better trial wave function?i. ...ii. ...Explain. Get solution
4. A particle of mass m is in the one-dimensional potential given by V(x) = Kx3 for x ≥ 0, where K is a positive constant. There is an infinite potential barrier at x = 0, so V(0) = ∞. Use the variational principle with the trial wave function ∣ψ〉 = xe−ax to estimate the ground-state energy. Get solution
5. Repeat the calculation in Example 10.1 using the trial wave function...where α is the parameter to be varied. Is the final result a better or a worse approximation to the true ground-state energy than the result of Example 10.1? Get solution
6. (a) A particle of mass m is in the one-dimensional potential given by V(x) = Kx4, where K is a positive constant. Use the variational principle with the trial wave function ... to estimate the ground-state energy.(b) The true ground-state wave function for this potential is a symmetric function of x, i.e., ψ0(−x) = ψ0(x). Use the result of Exercise 10.2, along with an appropriately chosen trial wave function, to estimate the energy of the first excited state. Get solution
7. A three-dimensional spherically-symmetric harmonic oscillator has the potential V(r) = (1/2)Kr2. The full Hamiltonian is then...[Note that the L2 operator has been written out in terms of derivatives.](a) Use the trial wave function ψ(r) = e−αr to calculate an approximation to the ground-state energy of the harmonic oscillator.(b) The exact ground-state energy for the three-dimensional harmonic oscillator is E = (3/2)ħω. What is the relative error in the estimate from part (a)? Get solution
8. Here is another approach to solve for the ground-state energy of helium.(a) Begin with the Hamiltonian of Equation (10.5), but neglect the interaction between the two electrons. Solve the Schrödinger equation in this case to derive the wave function of the two electrons and the energy.(b) Now add the interaction of the electrons as a perturbation:...Use first-order perturbation theory to calculate the change in energy, and add this change to the energy derived in part (a) to give an estimate for the total ground-state energy.(c) Is the estimate in part (b) more accurate or less accurate than the estimate from the variational principle? Get solution
9. (a) Singly-ionized lithium has a nucleus of charge +3e and two electrons. Use the variational principle to estimate the ground-state energy.(b) Now consider a nucleus of charge Ze with two electrons. Use the variational principle to estimate the ground-state energy. Get solution
2. Suppose that the trial wave function ∣ψ〉 used in the variational principle is orthogonal to the ground-state wave function of the Hamiltonian: 〈ψ0∣ψ(α)〉 = 0 for all values of α. Show that in this case...where E1 is the energy of the first excited state of H. Get solution
3. (a) In order to use the variational principle to estimate the ground-state energy of the one-dimensional potential V(x) = Kx4, where K is a constant, which of the following wave functions would be a better trial wave function?i. ...ii. ...Explain.(b) In order to use the variational principle to estimate the ground-state energy of the one-dimensional potential V(x) = Kx3 for x > 0 with an infinite potential barrier at x = 0, which of the following wave functions would be a better trial wave function?i. ...ii. ...Explain. Get solution
4. A particle of mass m is in the one-dimensional potential given by V(x) = Kx3 for x ≥ 0, where K is a positive constant. There is an infinite potential barrier at x = 0, so V(0) = ∞. Use the variational principle with the trial wave function ∣ψ〉 = xe−ax to estimate the ground-state energy. Get solution
5. Repeat the calculation in Example 10.1 using the trial wave function...where α is the parameter to be varied. Is the final result a better or a worse approximation to the true ground-state energy than the result of Example 10.1? Get solution
6. (a) A particle of mass m is in the one-dimensional potential given by V(x) = Kx4, where K is a positive constant. Use the variational principle with the trial wave function ... to estimate the ground-state energy.(b) The true ground-state wave function for this potential is a symmetric function of x, i.e., ψ0(−x) = ψ0(x). Use the result of Exercise 10.2, along with an appropriately chosen trial wave function, to estimate the energy of the first excited state. Get solution
7. A three-dimensional spherically-symmetric harmonic oscillator has the potential V(r) = (1/2)Kr2. The full Hamiltonian is then...[Note that the L2 operator has been written out in terms of derivatives.](a) Use the trial wave function ψ(r) = e−αr to calculate an approximation to the ground-state energy of the harmonic oscillator.(b) The exact ground-state energy for the three-dimensional harmonic oscillator is E = (3/2)ħω. What is the relative error in the estimate from part (a)? Get solution
8. Here is another approach to solve for the ground-state energy of helium.(a) Begin with the Hamiltonian of Equation (10.5), but neglect the interaction between the two electrons. Solve the Schrödinger equation in this case to derive the wave function of the two electrons and the energy.(b) Now add the interaction of the electrons as a perturbation:...Use first-order perturbation theory to calculate the change in energy, and add this change to the energy derived in part (a) to give an estimate for the total ground-state energy.(c) Is the estimate in part (b) more accurate or less accurate than the estimate from the variational principle? Get solution
9. (a) Singly-ionized lithium has a nucleus of charge +3e and two electrons. Use the variational principle to estimate the ground-state energy.(b) Now consider a nucleus of charge Ze with two electrons. Use the variational principle to estimate the ground-state energy. Get solution
