1. The electron in a hydrogen atom is initially in the ground state.
At t = 0, a homogeneous electric field aligned in the z direction is
turned on. The magnitude of the electric field decreases
exponentially:...where ℰ0 and τ are constants. A measurement is made at
tf = + ∞; what is the probability that the electron will be in the first
excited state? Get solution
2. A system is in an eigenstate |ψi〉 with energy Ei. The perturbation...is turned on at ti = −∞ and left on until tf = + ∞. Here ... is independent of time, and α is a constant. Show that at tf = + ∞, the probability that the system has evolved into the eigenstate |ψf〉 with energy Ef is... Get solution
3. An electron is in a strong, uniform, constant magnetic field with magnitude B0 aligned in the +x direction. The electron is initially in the state | →〉 with x component of spin equal to +ħ/2. A weak, uniform, constant magnetic field of magnitude B1(where B1≪ B0) in the +z direction is turned on at t = 0 and turned off at t = t0. Let P(i → f) be the probability that the electron is in the state | ←〉 with x component of spin equal to −ħ/2 at a later time tf > t0. Show that... Get solution
4. Consider a time-dependent perturbation H1(t) which is adiabatic, i.e., slowly varying. The system is initially in the state |ψi〉 at ti = −∞. The potential is turned on, and we wish to derive the probability that the system will be in the state |ψf〉 t some later time tf. Write down the standard expression for cf in this case and use integration by parts to break the expression into two terms, one of which contains dH1/dt. Since H1 is slowly varying, this term may be taken to be 0. Then use the fact that H1(−∞) = 0 to derive the final expression for the transition probability:... Get solution
5. An electron is in a strong, static, homogeneous magnetic field with magnitude B0 in the z direction. At time t = 0, the spin of the electron is in the +z direction. At t = 0 a weak, homogenous magnetic field with magnitude B1(where B1 ≪ B0) is turned on. At t = 0 this field is pointing in the x direction, but it rotates counterclockwise in the x-z plane with angular frequency ω, so that at any later time t this field is at an angle ωt relative to the x-axis:...Calculate the probability that at a later time tf the electron spin has flipped to the −z direction. Get solution
6. A particle with mass m is in a one-dimensional infinite square-well potential of width a, so V(x) = 0 for 0 ≤ x ≤ a, and there are infinite potential barriers at x = 0 and x = a. Recall that the normalized solutions to the Schrödinger equation are...with energies...where n = 1, 2, 3,…The particle is initially in the ground state. A delta-function potential...(where K is a constant) is turned on at time t = −t1 and turned off at t = t1. A measurement is made at some later time t2, where t2 > t1.(a) What is the probability that the particle will be found to be in the excited state n = 3?(b) There are some excited states n in which the particle will never be found, no matter what values are chosen for t1 and t2. Which excited states are these? Get solution
7. A hydrogen atom is in the ground state. At t = 0 an electric field with magnitude ℰ is turned on. At t = 0 the electric field points in the x direction, and it rotates counterclockwise in the x-y plane with angular frequency ω (i.e., at any later time t the field is oriented at an angle ωt relative to the x-axis). This rotating field causes the atom to undergo a transition to an n = 2 state. Determine which of the l, ml states are possible final states and which are impossible. Get solution
8. (a) Consider an electromagnetic wave polarized in the x direction, incident on a hydrogen atom. Show that in this case, the selection rules for ml are...(b) Repeat this calculation for an electromagnetic wave polarized in the y direction. Get solution
9. A hydrogen atom in the n = 4 state emits electric dipole radiation and drops into the n = 3 state. Determine all possible transitions in terms of their initial and final values for l and j. Express the answer in spectroscopic notation. Get solution
10. (a) The electron in a hydrogen atom is initially in the state n = 5, l = 0, j = 1/2. The atom emits electric dipole radiation and drops into an n = 3 state. Determine all l, j states which are possible final states.(b) An electron in a hydrogen atom is initially in the state n = 5, l = 2, j = 5/2. It emits electric dipole radiation and drops into the state n = 4, l = l1, j = j1. From this state, it emits electric dipole radiation again and drops into the hydrogen ground state. Determine l1 and j1. Get solution
11. An electron is contained in a three-dimensional rectangular box given by 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c. The solutions of the Schrödinger equation are specified by the quantum numbers nx, ny, and nz. Recall that the normalized wave function is...with energy...where nx, ny, and nz are positive integers. The electron is initially in the state nx, ny, nz. An electromagnetic wave is incident polarized in the y direction, so that the electric field vector is given by:...where E0 is a constant vector in the y direction. Use the dipole approximation and time-dependent perturbation theory to derive the selection rules for the electron to absorb the radiation and end up in the final state .... Get solution
2. A system is in an eigenstate |ψi〉 with energy Ei. The perturbation...is turned on at ti = −∞ and left on until tf = + ∞. Here ... is independent of time, and α is a constant. Show that at tf = + ∞, the probability that the system has evolved into the eigenstate |ψf〉 with energy Ef is... Get solution
3. An electron is in a strong, uniform, constant magnetic field with magnitude B0 aligned in the +x direction. The electron is initially in the state | →〉 with x component of spin equal to +ħ/2. A weak, uniform, constant magnetic field of magnitude B1(where B1≪ B0) in the +z direction is turned on at t = 0 and turned off at t = t0. Let P(i → f) be the probability that the electron is in the state | ←〉 with x component of spin equal to −ħ/2 at a later time tf > t0. Show that... Get solution
4. Consider a time-dependent perturbation H1(t) which is adiabatic, i.e., slowly varying. The system is initially in the state |ψi〉 at ti = −∞. The potential is turned on, and we wish to derive the probability that the system will be in the state |ψf〉 t some later time tf. Write down the standard expression for cf in this case and use integration by parts to break the expression into two terms, one of which contains dH1/dt. Since H1 is slowly varying, this term may be taken to be 0. Then use the fact that H1(−∞) = 0 to derive the final expression for the transition probability:... Get solution
5. An electron is in a strong, static, homogeneous magnetic field with magnitude B0 in the z direction. At time t = 0, the spin of the electron is in the +z direction. At t = 0 a weak, homogenous magnetic field with magnitude B1(where B1 ≪ B0) is turned on. At t = 0 this field is pointing in the x direction, but it rotates counterclockwise in the x-z plane with angular frequency ω, so that at any later time t this field is at an angle ωt relative to the x-axis:...Calculate the probability that at a later time tf the electron spin has flipped to the −z direction. Get solution
6. A particle with mass m is in a one-dimensional infinite square-well potential of width a, so V(x) = 0 for 0 ≤ x ≤ a, and there are infinite potential barriers at x = 0 and x = a. Recall that the normalized solutions to the Schrödinger equation are...with energies...where n = 1, 2, 3,…The particle is initially in the ground state. A delta-function potential...(where K is a constant) is turned on at time t = −t1 and turned off at t = t1. A measurement is made at some later time t2, where t2 > t1.(a) What is the probability that the particle will be found to be in the excited state n = 3?(b) There are some excited states n in which the particle will never be found, no matter what values are chosen for t1 and t2. Which excited states are these? Get solution
7. A hydrogen atom is in the ground state. At t = 0 an electric field with magnitude ℰ is turned on. At t = 0 the electric field points in the x direction, and it rotates counterclockwise in the x-y plane with angular frequency ω (i.e., at any later time t the field is oriented at an angle ωt relative to the x-axis). This rotating field causes the atom to undergo a transition to an n = 2 state. Determine which of the l, ml states are possible final states and which are impossible. Get solution
8. (a) Consider an electromagnetic wave polarized in the x direction, incident on a hydrogen atom. Show that in this case, the selection rules for ml are...(b) Repeat this calculation for an electromagnetic wave polarized in the y direction. Get solution
9. A hydrogen atom in the n = 4 state emits electric dipole radiation and drops into the n = 3 state. Determine all possible transitions in terms of their initial and final values for l and j. Express the answer in spectroscopic notation. Get solution
10. (a) The electron in a hydrogen atom is initially in the state n = 5, l = 0, j = 1/2. The atom emits electric dipole radiation and drops into an n = 3 state. Determine all l, j states which are possible final states.(b) An electron in a hydrogen atom is initially in the state n = 5, l = 2, j = 5/2. It emits electric dipole radiation and drops into the state n = 4, l = l1, j = j1. From this state, it emits electric dipole radiation again and drops into the hydrogen ground state. Determine l1 and j1. Get solution
11. An electron is contained in a three-dimensional rectangular box given by 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c. The solutions of the Schrödinger equation are specified by the quantum numbers nx, ny, and nz. Recall that the normalized wave function is...with energy...where nx, ny, and nz are positive integers. The electron is initially in the state nx, ny, nz. An electromagnetic wave is incident polarized in the y direction, so that the electric field vector is given by:...where E0 is a constant vector in the y direction. Use the dipole approximation and time-dependent perturbation theory to derive the selection rules for the electron to absorb the radiation and end up in the final state .... Get solution
