Chapter 8 Quiz

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 8 Quiz

Instructions: Select the best answer for each question. Answers and explanations follow at the end.

Q1. The ket $|\psi\rangle$ represents:

(a) The wave function $\psi(x)$ (b) An abstract state vector in Hilbert space, independent of any basis (c) The probability amplitude for finding the particle at position $x$ (d) The column vector of expansion coefficients in the energy basis

Q2. The wave function $\psi(x)$ is related to the ket $|\psi\rangle$ by:

Q3. If $|\psi\rangle = (2 + i)|a\rangle + 3|b\rangle$ where $\{|a\rangle, |b\rangle\}$ is orthonormal, then $\langle\psi|$ is:

Q4. The inner product $\langle\phi|\psi\rangle$ satisfies:

Q5. The completeness relation $\sum_n |n\rangle\langle n| = \hat{I}$ means:

(a) The basis states are orthonormal (b) The basis states span the entire Hilbert space (c) The sum of eigenvalues is 1 (d) The operator $\hat{I}$ has eigenvalue 1

Q6. The primary computational technique introduced in this chapter is:

(a) Separation of variables (b) Power series solution (c) Inserting a complete set of states (d) Integration by parts

Q7. The matrix element $A_{mn} = \langle m|\hat{A}|n\rangle$ is:

(a) Always real (b) Always positive (c) The $(m, n)$ entry of the matrix representing $\hat{A}$ in the basis $\{|n\rangle\}$ (d) The eigenvalue of $\hat{A}$ corresponding to state $|n\rangle$

Q8. The outer product $|n\rangle\langle m|$ is:

(a) A complex number (b) A ket (c) A bra (d) An operator

Q9. The projection operator $\hat{P}_n = |n\rangle\langle n|$ satisfies which of the following?

(a) $\hat{P}_n^2 = 2\hat{P}_n$ (b) $\hat{P}_n^2 = \hat{P}_n$ (c) $\hat{P}_n^2 = \hat{I}$ (d) $\hat{P}_n^2 = 0$

Q10. The Hermitian conjugate of $\hat{A}\hat{B}$ is:

(a) $\hat{A}^\dagger\hat{B}^\dagger$ (b) $\hat{B}^\dagger\hat{A}^\dagger$ (c) $\hat{B}\hat{A}$ (d) $(\hat{A}\hat{B})^T$

Q11. For the quantum harmonic oscillator, $\langle n|\hat{x}|n\rangle = 0$ because:

(a) The position operator is anti-Hermitian (b) The state $|n\rangle$ has zero energy (c) The ladder operators change $n$ by $\pm 1$, and orthogonality kills the diagonal terms (d) The Hamiltonian commutes with $\hat{x}$

Q12. A unitary operator $\hat{U}$ satisfies:

(a) $\hat{U}^\dagger = \hat{U}$ (b) $\hat{U}^\dagger = -\hat{U}$ (c) $\hat{U}^\dagger\hat{U} = \hat{I}$ (d) $\hat{U}^2 = \hat{I}$

Q13. The Fourier transform connecting $\psi(x)$ and $\phi(p)$ is, in Dirac notation:

(a) An eigenvalue equation (b) A change of basis (unitary transformation) (c) A symmetry operation (d) A projection

Q14. The trace of an operator is:

(a) Always zero for Hermitian operators (b) Dependent on the choice of basis (c) Independent of the choice of basis (d) Equal to the largest eigenvalue

Q15. The cyclic property of the trace states that:

(a) $\text{Tr}(\hat{A}\hat{B}) = \text{Tr}(\hat{A})\text{Tr}(\hat{B})$ (b) $\text{Tr}(\hat{A}\hat{B}\hat{C}) = \text{Tr}(\hat{C}\hat{A}\hat{B})$ (c) $\text{Tr}(\hat{A}\hat{B}) = \text{Tr}(\hat{A}) + \text{Tr}(\hat{B})$ (d) $\text{Tr}(\hat{A}\hat{B}) = 0$ if $[\hat{A}, \hat{B}] \neq 0$

Q16. In the $S_z$ eigenbasis, the state $|\uparrow\rangle$ expressed in the $S_x$ eigenbasis is:

(a) $|+x\rangle$ (b) $\frac{1}{\sqrt{2}}(|+x\rangle + |-x\rangle)$ (c) $\frac{1}{\sqrt{2}}(|+x\rangle - |-x\rangle)$ (d) $|-x\rangle$

Q17. The spectral decomposition $\hat{A} = \sum_n a_n |a_n\rangle\langle a_n|$ allows us to define $f(\hat{A})$ as:

(a) $f(\hat{A}) = f(a_1)\hat{I}$ (b) $f(\hat{A}) = \sum_n f(a_n) |a_n\rangle\langle a_n|$ (c) $f(\hat{A}) = f\left(\sum_n a_n\right) \hat{I}$ (d) $f(\hat{A}) = \sum_n a_n f(|a_n\rangle\langle a_n|)$

Q18. A student claims that "$|\psi\rangle$ and $\psi(x)$ are the same thing." The best response is:

(a) That is correct — they are different notations for the same object. (b) That is incorrect — $|\psi\rangle$ is the abstract state, while $\psi(x) = \langle x|\psi\rangle$ is its position representation. (c) That is incorrect — $|\psi\rangle$ exists only for finite-dimensional systems. (d) That is incorrect — $\psi(x)$ is more fundamental than $|\psi\rangle$.

Q19. For a spin-1/2 system, the dimension of the Hilbert space is:

(a) 1 (b) 2 (c) 3 (d) Infinite

Q20. Which of the following is NOT a valid operation?

(a) $\langle\phi|\psi\rangle$ (inner product giving a scalar) (b) $|\phi\rangle\langle\psi|$ (outer product giving an operator) (c) $|\phi\rangle|\psi\rangle$ (two kets multiplied — not defined within a single Hilbert space) (d) $\hat{A}|\psi\rangle$ (operator acting on a ket)

Q21. The expression $\langle x|\hat{p}|\psi\rangle = -i\hbar\frac{\partial}{\partial x}\langle x|\psi\rangle$ tells us that:

(a) The momentum operator is always $-i\hbar\frac{d}{dx}$, regardless of representation (b) In the position representation, the momentum operator acts as a derivative (c) Position and momentum cannot be measured simultaneously (d) The momentum wave function is the derivative of the position wave function

Q22. The completeness relation for position eigenstates is $\int |x\rangle\langle x| \, dx = \hat{I}$. The analogous relation for momentum eigenstates is:

(a) $\int |p\rangle\langle p| \, dp = \hat{I}$ (b) $\sum_n |p_n\rangle\langle p_n| = \hat{I}$ (c) $\int |p\rangle\langle p| \, dp = 2\pi\hbar\hat{I}$ (d) $|p\rangle\langle p| = \hat{I}$

Answer Key

Q1: (b) The ket is an abstract vector in Hilbert space. The wave function, column vector, and probability amplitude are all representations of this abstract object in specific bases.

Q2: (c) This is the fundamental relation connecting the abstract ket to its position representation. The wave function is the projection of the ket onto a position eigenstate.

Q3: (d) Both (b) and (c) say the same thing. When converting ket to bra, complex conjugate all scalars: $(2+i)^* = 2-i$ and $3^* = 3$.

Q4: (c) Conjugate symmetry is a fundamental property of the inner product. It follows from $\langle\phi|\psi\rangle = \int \phi^*(x)\psi(x) \, dx$ and the conjugation of the integral.

Q5: (b) Completeness means that any state in the Hilbert space can be expanded in the basis. Orthonormality ($\langle m|n\rangle = \delta_{mn}$) is a separate condition. Together, orthonormality and completeness define a complete orthonormal basis.

Q6: (c) Inserting a complete set of states ($\sum_n |n\rangle\langle n| = \hat{I}$ or $\int |x\rangle\langle x| \, dx = \hat{I}$) is the fundamental technique that enables basis changes, matrix element evaluations, and translations between representations.

Q7: (c) This is the definition of the matrix representation of an operator. The matrix element is a complex number in general.

Q9: (b) This is the defining property of a projection operator (idempotency): $\hat{P}_n^2 = |n\rangle\langle n|n\rangle\langle n| = |n\rangle\langle n| = \hat{P}_n$, using $\langle n|n\rangle = 1$.

Q10: (b) The Hermitian conjugate reverses the order: $(\hat{A}\hat{B})^\dagger = \hat{B}^\dagger\hat{A}^\dagger$. This is analogous to $(AB)^T = B^T A^T$ for matrix transposition.

Q11: (c) $\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}(\hat{a} + \hat{a}^\dagger)$. The lowering operator gives $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$ and the raising operator gives $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$. Both produce states orthogonal to $|n\rangle$, so $\langle n|\hat{a}|n\rangle = \langle n|\hat{a}^\dagger|n\rangle = 0$.

Q12: (c) This is the definition of a unitary operator. Note: (a) defines a Hermitian operator, and (d) defines an involution (which may or may not be unitary).

Q13: (b) The Fourier transform is a unitary change of basis from the position representation to the momentum representation. The "basis change matrix" is $\langle p|x\rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{-ipx/\hbar}$.

Q14: (c) The trace is basis-independent, as proven in Section 8.7 using the completeness relation. This is one of its most important properties.

Q15: (b) The cyclic property allows cyclic permutations of operators under the trace. Note that this does NOT mean arbitrary permutations: $\text{Tr}(\hat{A}\hat{B}\hat{C}) \neq \text{Tr}(\hat{A}\hat{C}\hat{B})$ in general.

Q16: (b) Since $|+x\rangle = \frac{1}{\sqrt{2}}(|\uparrow\rangle + |\downarrow\rangle)$ and $|-x\rangle = \frac{1}{\sqrt{2}}(|\uparrow\rangle - |\downarrow\rangle)$, inverting gives $|\uparrow\rangle = \frac{1}{\sqrt{2}}(|+x\rangle + |-x\rangle)$.

Q17: (b) This is the definition of a function of an operator via spectral decomposition: replace each eigenvalue $a_n$ with $f(a_n)$.

Q18: (b) This is the threshold concept of the chapter. The ket is the abstract state; the wave function is one particular representation.

Q19: (b) A spin-1/2 particle has two basis states ($|\uparrow\rangle$ and $|\downarrow\rangle$), so its Hilbert space is two-dimensional. This is why spin can be described by $2 \times 2$ matrices.

Q20: (c) Within a single Hilbert space, the "product" of two kets $|\phi\rangle|\psi\rangle$ is not defined. (In the context of composite systems, Chapter 11, this becomes a tensor product $|\phi\rangle \otimes |\psi\rangle$, which lives in a larger Hilbert space.)

Q21: (b) The expression tells us how $\hat{p}$ acts in the position representation specifically. In the momentum representation, $\hat{p}$ acts by simple multiplication: $\langle p|\hat{p}|\psi\rangle = p\langle p|\psi\rangle$. The operator itself is abstract; its form depends on the representation.

Q22: (a) Both position and momentum eigenstates form complete bases with the same structure. The factors of $2\pi\hbar$ are absorbed into the normalization convention $\langle p|p'\rangle = \delta(p - p')$.