1. Verify the commutator properties given in Equations (5.4)–(5.7), i.e., for any operators A, B, and C, show that... Get solution
2. Consider a particle moving in three dimensions. Is it possible for the particle to be in a state of definite px and y, i.e., can both its y-coordinate and its momentum in the x direction be known at the same time? Get solution
3. (a) Verify that the ordinary dot product for three-dimensional vectors satisfies all of the properties of an inner product, given by Equations (5.11)–(5.14).(b) Verify that the inner product for complex-valued, three-dimensional functions defined in Equation (5.16) satisfies Equations (5.11)–(5.14). Get solution
4. (a) The operators A, B, and C are all Hermitian with [A, B] = C. Show that C = 0.(b) The operators A and B are both Hermitian with [A, B] = iħ. Determine whether or not AB is a Hermitian operator. Get solution
5. The one-dimensional parity operator Π is defined by Πψ(x) = ψ(−x). In other words, n changes x into −x everywhere in the function.(a) Is Π a Hermitian operator?(b) For what potentials V(x) is it possible to find a set of wavefunctions which are eigenfunctions of the parity operator and solutions of the one-dimensional time-independent Schrödinger equation? Get solution
6. (a) Let Q be an operator which is not a function of time, and let H be the Hamiltonian operator. Show that...Here 〈q〉 is the expectation value of Q for an arbitrary time-dependent wave function Ψ, which is not necessarily an eigenfunction of H, and 〈[Q, H]〉 is the expectation value of the commutator of Q and H for the same wave function. This result is known as Ehrenfest’s theorem.(b) Use this result to show that...What is the classical analog of this equation? Get solution
7. (a) Show that the one-dimensional momentum operator is Hermitian.(b) Use this result to show that the one-dimensional Hamiltonian operator H with potential V(x) is Hermitian. What (reasonable) assumption must be made about V(x) to derive this result? Get solution
8. Suppose that the operator T is defined by T = αQ†Q, where or is a real number, and Q is an operator (not necessarily Hermitian). Show that T is Hermitian. Get solution
9. Determine all potentials V(x) for which it is possible to find a set of solutions of the time-independent Schrödinger equation which are also eigenfunctions of the position operator X, or else show that no such potentials exist. Get solution
10. Suppose that two operators P and Q satisfy the commutation relation...Suppose that ψ is an eigenfunction of the operator P with eigenvalue p. Show that Qψ is also an eigenfunction of P, and find its eigenvalue. Get solution
11. The operator F is defined by Fψ(x) = ψ(x + a) + ψ(x − a), where a is a nonzero constant. Determine whether or not F is a Hermitian operator. Get solution
2. Consider a particle moving in three dimensions. Is it possible for the particle to be in a state of definite px and y, i.e., can both its y-coordinate and its momentum in the x direction be known at the same time? Get solution
3. (a) Verify that the ordinary dot product for three-dimensional vectors satisfies all of the properties of an inner product, given by Equations (5.11)–(5.14).(b) Verify that the inner product for complex-valued, three-dimensional functions defined in Equation (5.16) satisfies Equations (5.11)–(5.14). Get solution
4. (a) The operators A, B, and C are all Hermitian with [A, B] = C. Show that C = 0.(b) The operators A and B are both Hermitian with [A, B] = iħ. Determine whether or not AB is a Hermitian operator. Get solution
5. The one-dimensional parity operator Π is defined by Πψ(x) = ψ(−x). In other words, n changes x into −x everywhere in the function.(a) Is Π a Hermitian operator?(b) For what potentials V(x) is it possible to find a set of wavefunctions which are eigenfunctions of the parity operator and solutions of the one-dimensional time-independent Schrödinger equation? Get solution
6. (a) Let Q be an operator which is not a function of time, and let H be the Hamiltonian operator. Show that...Here 〈q〉 is the expectation value of Q for an arbitrary time-dependent wave function Ψ, which is not necessarily an eigenfunction of H, and 〈[Q, H]〉 is the expectation value of the commutator of Q and H for the same wave function. This result is known as Ehrenfest’s theorem.(b) Use this result to show that...What is the classical analog of this equation? Get solution
7. (a) Show that the one-dimensional momentum operator is Hermitian.(b) Use this result to show that the one-dimensional Hamiltonian operator H with potential V(x) is Hermitian. What (reasonable) assumption must be made about V(x) to derive this result? Get solution
8. Suppose that the operator T is defined by T = αQ†Q, where or is a real number, and Q is an operator (not necessarily Hermitian). Show that T is Hermitian. Get solution
9. Determine all potentials V(x) for which it is possible to find a set of solutions of the time-independent Schrödinger equation which are also eigenfunctions of the position operator X, or else show that no such potentials exist. Get solution
10. Suppose that two operators P and Q satisfy the commutation relation...Suppose that ψ is an eigenfunction of the operator P with eigenvalue p. Show that Qψ is also an eigenfunction of P, and find its eigenvalue. Get solution
11. The operator F is defined by Fψ(x) = ψ(x + a) + ψ(x − a), where a is a nonzero constant. Determine whether or not F is a Hermitian operator. Get solution
