QM2 Concept Test 10.1 Consider the Hamiltonian 𝐻 0 + 𝐻 ′= 𝑉 0 1−𝜀 𝜀 0 𝜀 1 𝜀 0 𝜀 2 , where 𝜀≪1. The basis vectors for the matrix 𝑎 , 𝑏 , and 𝑐 are the energy eigenstates of the unperturbed Hamiltonian 𝐻 0 (𝜀=0). Choose all of the following statements that are correct. 𝑎 is a “good” state for the perturbation 𝐻 ′. 𝑐 is a “good” state for the perturbation 𝐻 ′. The perturbation matrix in the degenerate subspace of 𝐻 0 is 𝑉 0 −𝜀 𝜀 𝜀 0 . A. 1 only B. 2 only C. 1 and 3 only D. 2 and 3 only E. All of the above

QM2 Concept Test 10.2 Consider the Hamiltonian 𝐻 0 + 𝐻 ′= 𝑉 0 1−𝜀 𝜀 0 𝜀 1 𝜀 0 𝜀 2 , where 𝜀≪1. The basis vectors for the matrix 𝑎 , 𝑏 , and 𝑐 are the energy eigenstates of the unperturbed Hamiltonian 𝐻 0 (𝜀=0). Choose all of the following statements that are correct. The first order correction to the energy for the state 𝑎 is −𝜀 𝑉 0 . The first order correction to the energy for the state 𝑏 is zero. The first order correction to the energy for the state 𝑐 is zero. A. 1 only B. 3 only C. 1 and 2 only D. 1 and 3 only E. All of the above.

QM2 Concept Test 10.3 A perturbation 𝐻 ′ acts on a hydrogen atom with the unperturbed Hamiltonian 𝐻 0 =− ℏ 2 2𝑚 𝛻 2 − 𝑒 2 4𝜋 𝜀 0 1 𝑟 . To calculate the perturbative corrections, we use 𝑛,𝑙, 𝑚 𝑙 , 𝑠, 𝑚 𝑠 (the eigenstates of ( 𝐻 0 , 𝐿 2 , and 𝐿 𝑧 ) as the basis vectors. Choose all of the following statements that are correct. If 𝐻 ′ =𝛼 𝐿 𝑧 , where 𝛼 is a suitable constant, we can calculate the first order corrections as 𝐸 1 = 𝑛,𝑙, 𝑚 𝑙 , 𝑠, 𝑚 𝑠 𝐻 ′ 𝑛,𝑙, 𝑚 𝑙 , 𝑠, 𝑚 𝑠 . If 𝐻 ′ =𝛼𝛿(𝑟), the first order correction to energy is 𝐸 1 = 𝑛,𝑙, 𝑚 𝑙 , 𝑠, 𝑚 𝑠 𝐻 ′ 𝑛,𝑙, 𝑚 𝑙 , 𝑠, 𝑚 𝑠 . If 𝐻 ′ =𝛼 𝐽 𝑧 (𝑧 component of 𝐽 = 𝐿 + 𝑆 ) we can calculate the first order correction as 𝐸 1 = 𝑛,𝑙, 𝑚 𝑙 , 𝑠, 𝑚 𝑠 𝐻 ′ 𝑛,𝑙, 𝑚 𝑙 , 𝑠, 𝑚 𝑠 . A. 1 only B. 2 only C. 1 and 2 only D. 1 and 3 only E. All of the above

QM2 Concept Test 10.4 A perturbation 𝐻 ′ acts on a hydrogen atom with the unperturbed Hamiltonian 𝐻 0 =− ℏ 2 2𝑚 𝛻 2 − 𝑒 2 4𝜋 𝜀 0 1 𝑟 . To calculate the perturbative corrections, we use the coupled representation 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 as the basis vectors. Choose all of the following statements that are correct. If 𝐻 ′ =𝛼 𝐿 𝑧 , where 𝛼 is a suitable constant, we can calculate the first order corrections as 𝐸 1 = 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 𝐻 ′ 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 . If 𝐻 ′ =𝛼𝛿(𝑟), the first order correction to energy is 𝐸 1 = 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 𝐻 ′ 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 . If 𝐻 ′ =𝛼 𝐽 𝑧 (𝑧 component of 𝐽 = 𝐿 + 𝑆 ) we can calculate the first order correction as 𝐸 1 = 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 𝐻 ′ 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 . A. 1 only B. 1 and 2 only C. 1 and 3 only D. 2 and 3 only E. All of the above

QM2 Concept Test 10.5 A perturbation 𝐻 ′ acts on a hydrogen atom with the unperturbed Hamiltonian 𝐻 0 =− ℏ 2 2𝑚 𝛻 2 − 𝑒 2 4𝜋 𝜀 0 1 𝑟 . Choose all of the following statements that are correct. If 𝐻 ′ =𝛼( 𝐿 𝑧 + 𝑆 𝑧 ) we can calculate the first order correction as 𝐸 1 = 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 𝐻 ′ 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 . If 𝐻 ′ =𝛼( 𝐿 𝑧 + 𝑆 𝑧 2 ) we can calculate the first order correction as 𝐸 1 = 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 𝐻 ′ 𝑛,𝑙,𝑠,𝑗, 𝑚 𝑗 . If 𝐻 ′ =𝛼( 𝐿 𝑧 + 𝑆 𝑧 2 ) we can calculate the first order correction as 𝐸 1 = 𝑛,𝑙, 𝑚 𝑙 , 𝑠, 𝑚 𝑠 𝐻 ′ 𝑛,𝑙, 𝑚 𝑙 , 𝑠, 𝑚 𝑠 . A. 1 only B. 1 and 2 only C. 1 and 3 only D. 2 and 3 only E. All of the above

QM2 Concept Test 10.6 Suppose the eigenstates 𝑎 , 𝑏 , and 𝑐 of 𝐻 0 are 3-fold degenerate with energy 𝐸 0 . A perturbation 𝐻 ’ acts on this system and the perturbation matrix is 𝐻 ′ =𝑉 1 0 0 0 1 1 0 1 1 if 𝑎 , 𝑏 , and 𝑐 are used as basis vectors. Note: The matrix 1 1 1 1 has eigenvectors 1 2 1 1 and 1 2 1 −1 with eigenvalues 2 and 0, respectively. Choose all of the following statements that are correct. The perturbation 𝐻 ′ splits the original energy level 𝐸 0 into three distinct energy levels because the eigenvalues of the matrix are 0, 1, and 2. 𝑎 is a “good” state for this system. 𝑏 + 𝑐 2 and 𝑏 − 𝑐 2 are “good” states for this system. A. 1 only B. 2 only C. 1 and 2 only D. 2 and 3 only E. All of the above.

QM2 Concept Test 10.7 Suppose the unperturbed Hamiltonian 𝐻 0 is two-fold degenerate, i.e., 𝐻 0 𝜓 𝑎 0 = 𝐸 1 0 𝜓 𝑎 0 , 𝐻 0 𝜓 𝑏 0 = 𝐸 1 0 𝜓 𝑏 0 , 𝜓 𝑎 0 𝜓 𝑏 0 =0. A perturbation 𝐻 ′ acts on this system and a Hermitian operator 𝐴 commutes with both 𝐻 0 and 𝐻 ′ . Choose all of the following statements that are correct. 𝐻 0 and 𝐻 ′ must commute with each other. If 𝜓 𝑎 0 and 𝜓 𝑏 0 are degenerate eigenstates of 𝐴 , they must be “good” states for finding perturbative corrections to the energy and wavefunction due to 𝐻 ′ . If 𝜓 𝑎 0 and 𝜓 𝑏 0 are non-degenerate eigenstates of 𝐴 , they must be “good” states. A. 1 only B. 2 only C. 3 only D. 1 and 2 only E. 1 and 3 only