1. The operator Q is given by the matrix:...(a) Determine the matrix
corresponding to Q†.(b) Is Q Hermitian?(c) Find the eigenvalues of
Q.(d) For each eigenvalue in part (c), determine the corresponding
eigenvector. Get solution
2. The operator A is given by the matrix:...(a) Is A Hermitian?(b) Find the eigenvalues and corresponding eigenvectors.(c) What is unusual about the eigenvectors corresponding to the eigenvalue c = 0? Get solution
3. Suppose that an n × n matrix A is diagonal so that Aij = 0 for i ≠ j, but the diagonal elements A11, A22, ... need not be zero. Find the eigenvalues and eigenvectors of this matrix. Get solution
4. The trace of a matrix A, written tr (A), is defined to be the sum of its diagonal elements:...(a) Show that for any two square matrices, tr(AB) = tr(BA).(b) Show that for any matrix A, the trace is equal to the sum of its eigenvalues (where multiple eigenvalues must be included in the sum multiple times). Get solution
5. Normalize these vectors: .... Get solution
6. A particle is in the state ∣ϕ〉, and let ∣ψ1〉, ∣ψ2〉, …, ∣ψn〉 be an orthonormal basis for the vector space which contains ∣ϕ〉. Q is a Hermitian operator. Show that... Get solution
7. Suppose that ∣ψ1〉, ∣ψ2〉, …, ∣ψn〉 is an orthonormal basis set, and all of the basis vectors are eigenvectors of the operator Q with Q∣ψn〉 = qj∣ψn〉 for all j. A particle is in the state ∣ϕ〉. Show that for this particle, the expectation value of Q is... Get solution
8. If the operator U has the property that U†U = 1 (where I is the identity operator), then U is called a unitary operator. Show that if ∣ψ1〉, ∣ψ2〉, …, ∣ψn〉 are a set of orthonormal vectors, then U∣ψ1〉, U∣ψ2〉, …, U∣ψn〉 are also a set of orthonormal vectors. Get solution
9. Suppose that U is a unitary operator, as defined in Exercise 7.8, and U is represented by a matrix. Show that the columns of U form a set of orthonormal column vectors. Get solution
10. A particle is in the state ∣ϕ〉. Let ∣ψ1〉, ∣ψ2〉, …, ∣ψn〉 be an orthonormal basis for the vector space which contains ∣ϕ〉, and assume that all of these basis vectors are eigenvectors of H with H∣ψj〉 = Ej∣ψj〉 for all j. Suppose that the operator Q satisfies...where Z is just the usual position operator in the z direction. Derive an expression for Q as a function of H and Z, but not containing Em. Get solution
11. The delta function is sometimes represented as...Show that this definition satisfies the properties of the delta function given in Equations (7.10)–(7.12). Get solution
12. Show that... Get solution
13. Show that...where x0 is determined by g(x0) = 0. Get solution
2. The operator A is given by the matrix:...(a) Is A Hermitian?(b) Find the eigenvalues and corresponding eigenvectors.(c) What is unusual about the eigenvectors corresponding to the eigenvalue c = 0? Get solution
3. Suppose that an n × n matrix A is diagonal so that Aij = 0 for i ≠ j, but the diagonal elements A11, A22, ... need not be zero. Find the eigenvalues and eigenvectors of this matrix. Get solution
4. The trace of a matrix A, written tr (A), is defined to be the sum of its diagonal elements:...(a) Show that for any two square matrices, tr(AB) = tr(BA).(b) Show that for any matrix A, the trace is equal to the sum of its eigenvalues (where multiple eigenvalues must be included in the sum multiple times). Get solution
5. Normalize these vectors: .... Get solution
6. A particle is in the state ∣ϕ〉, and let ∣ψ1〉, ∣ψ2〉, …, ∣ψn〉 be an orthonormal basis for the vector space which contains ∣ϕ〉. Q is a Hermitian operator. Show that... Get solution
7. Suppose that ∣ψ1〉, ∣ψ2〉, …, ∣ψn〉 is an orthonormal basis set, and all of the basis vectors are eigenvectors of the operator Q with Q∣ψn〉 = qj∣ψn〉 for all j. A particle is in the state ∣ϕ〉. Show that for this particle, the expectation value of Q is... Get solution
8. If the operator U has the property that U†U = 1 (where I is the identity operator), then U is called a unitary operator. Show that if ∣ψ1〉, ∣ψ2〉, …, ∣ψn〉 are a set of orthonormal vectors, then U∣ψ1〉, U∣ψ2〉, …, U∣ψn〉 are also a set of orthonormal vectors. Get solution
9. Suppose that U is a unitary operator, as defined in Exercise 7.8, and U is represented by a matrix. Show that the columns of U form a set of orthonormal column vectors. Get solution
10. A particle is in the state ∣ϕ〉. Let ∣ψ1〉, ∣ψ2〉, …, ∣ψn〉 be an orthonormal basis for the vector space which contains ∣ϕ〉, and assume that all of these basis vectors are eigenvectors of H with H∣ψj〉 = Ej∣ψj〉 for all j. Suppose that the operator Q satisfies...where Z is just the usual position operator in the z direction. Derive an expression for Q as a function of H and Z, but not containing Em. Get solution
11. The delta function is sometimes represented as...Show that this definition satisfies the properties of the delta function given in Equations (7.10)–(7.12). Get solution
12. Show that... Get solution
13. Show that...where x0 is determined by g(x0) = 0. Get solution
