Title: Part II / Principles of Quantum Mechanics
Tags: Perturbation::Theory Translations Introduction::to::Quantum::Mechanics Interpretation::of::Quantum::Mechanics Addition::of::Angular::Momentum Angular::Momentum Hilbert::Spaces Multiparticle

Q: What are the postulates of QM?
A: 1st: The states of a system are vectors in a Hilbert space
2nd: Observables quantities correspond to Hermitian linear operators
3rd: Time evolution obeys the Schrödinger equation

Q: What is Bra-Ket notation?
A: If \(\varphi \in \mathcal{H}\) we place it inside a "Ket", as \(\ket{\varphi}\)
If \(\phi \in \mathcal{H}^*\) we place it inside a "Bra", as \(\bra{\phi}\)
The inner product is a Bra-Ket

Q: What are the Leibnitz rule and Jacobi identity for the commutator of operators?
A: \[[A,BC] = [A,B]C + B[A,C]\]\[[A,[B,C]] + [B,[C,A]] + [C, [A,B]] = 0\]

Q: How is the adjoint of an operator defined?
A: The adjoint \(A^{\dagger}\) of \(A\) is the operator such that \[\bra{\phi} A^{\dagger} \ket{\varphi} = \bra{\varphi} A \ket{\phi}^* \; \; \forall \ket{\phi}, \ket{\varphi} \in \mathcal{H}\]

Q: What is the time-dependent Schrödinger equation?
A: \[i \hbar \frac{\partial}{\partial t} \ket{\varphi} = H \ket{\varphi}\]

Q: What is the uncertainty of a Hermitian operator in state \(\ket{\varphi}\)?
A: \[\Delta_{\varphi} Q = \sqrt{\langle Q^2 \rangle_{\varphi} - \langle Q\rangle_{\varphi}^2}\]

Q: What is \(\ket{\varphi_A}\) and why is it useful?
A: \[\ket{\varphi_A} = A \ket{\varphi} - \langle A \rangle_{\varphi} \ket{\varphi}\]\[|| \ket{\varphi_A}|| = \Delta_{\varphi}A\]

Q: What is the generalised uncertainty principle and how do you prove it?
A: Let \(\ket{\psi_A} = A\ket{\psi} - \langle A\rangle_{\psi}\ket{\psi}\) so that \((\Delta_{\psi} A)^2 = ||\ket{\psi_A}||^2\) and same for B.
Cauchy-Schwarz gives us \[(\Delta_{\psi}A)^2(\Delta_{\psi}B)^2 = ||\ket{\psi_A}||^2 ||\ket{\psi_B}||^2 \ge |\bra{\psi_A}\ket{\psi_B}|^2\]Expanding RHS we get \[\bra{\psi_A}\ket{\psi_B} = \bra{\psi}(A - \langle A \rangle_{\psi})(B - \langle B \rangle_{\psi})\ket{\psi} = \bra{\psi}AB-\langle A \rangle_{\psi} \langle B \rangle_{\psi} \ket{\psi}\]Since the operators are Hermitian we get \[\text{Im}\bra{\psi_A}\ket{\psi_B} = \frac{1}{2i} \bra{\psi}[A,B]\ket{\psi}\]Combining these we get \[(\Delta_{\psi}A)^2 (\Delta_{\psi}B)^2 \ge |\bra{\psi_A}\ket{\psi_B}|^2 \ge \frac{1}{4}|\langle [A,B]\rangle_{\psi}|^2\]

Q: What is the Hamiltonian for the 1d quantum harmonic oscillator, and what are the lowering and raising operators?
A: \[H = \frac{P^2}{2m} + \frac{m \omega^2}{2} X^2\]\[A = \frac{1}{\sqrt{2m\omega \hbar}} (m\omega X + i P), A^{\dagger} = \frac{1}{\sqrt{2m\omega \hbar}} (m\omega X - i P)\] And we get \[A^{\dagger}A = \frac{H}{\hbar \omega} - \frac{1}{2}\]

Q: What is the Fock Hilbert space?
A: \[\mathcal{H} = \text{Span}\{\ket{0}, \ket{1} , \ket{2} , \dots\}\]\[\dim \mathcal{H} = \infty\]The general state \(\ket{0}\) is defined as \(A\ket{0} = 0\)
Normalised energy H or number N eigenstates \[\ket{n} = \frac{1}{\sqrt{n!}} (A^{\dagger})^n \ket{0}\]Energy levels \[\bra{n}H \ket{n} = \hbar \omega (n + 1/2) \; \; \; n \in \mathbb{N}\]

Q: What does Wigner's Theorem say about transformations that conserve probabilities?
A: Transformations that conserve probabilities are represented by a linear and unitary operator \[\ket{\psi'} = U \ket{\psi} \; \; U^{\dagger} = U^{-1}\]
\[\text{Prob}(\phi \to \psi) = |\bra{\phi}\ket{\psi}|^2 = |\bra{\phi} U^{\dagger}U \ket{\psi}|^2 = |\bra{\phi'} \ket{\psi'}|^2 = \text{Prob}(\phi' \to \psi')\]

Q: How does the rotation matrix work and what is the commutator relation for the generators?
A: We have \(R(\mathbf{a}): \mathbf{v \to v' =}R(\mathbf{a})\mathbf{v}\)\[[J_i, J_j] = i\hbar \epsilon_{ijk} J_k\]

Q: What is the spin operator?
A: \[\mathbf{S = J - L}\]where \(\bf J\) generates roations and \(\bf L\) generates circular transformations

Q: Definition. How are the Heisenberg and Schrödinger picture operators and vectors related?
A: \[Q_H(t) = U^{\dagger}(t) Q_s U(t)\] where \(U(t) = e^{-iHt/\hbar}\) and \(Q_S = Q = Q_H(0)\)

Q: What is \(\frac{dQ_H(t)}{dt} \)?
A: \[= \frac{i}{\hbar} U^{\dagger}(t)[H,Q_s]U(t) + U^{\dagger}\frac{\partial Q_S}{\partial t} U(t) = \frac{i}{\hbar}[H, Q_H] + \frac{\partial Q_H}{\partial t} \big |_H\]

Q: Definition. What is a converved operator Q?
A: If \[\frac{dQ_H(t)}{dt} = 0\]For simplicity assume \(\partial_t Q_S = 0\), then in the Heisenberg picture \[\frac{dQ_H(t)}{dt} = \frac{i}{\hbar}[H, Q_H(t)]= \frac{i}{\hbar} U^{\dagger}(t)[H,Q]U(t)\]Since U is unitary, \(\frac{dQ_H(t)}{dt} = 0\) iff \([H,Q] = 0\)

Q: Definition. A transformation, \(U(\theta) = e^{-i\theta T}\), is a symmetry if [...]
A: it leaves H unchanged. \[U^{\dagger}HU = H \text{ or equivalently } [T,H] = 0\]

Q: How does the algebra \([J_i, J_j] = i\hbar\epsilon_{ijk}J_k\) determine the spectrum of the angular momentum operators?
A: \[\beta = j(j+1) \text{ where } 2j \in \mathbb{N}_0\]\[\mathcal{H}_j = \text{span}(\ket{j,m}) \text{ with } m \in \{-j, -j+1, \dots , j-1,j\}\]\[J_z\ket{j,m} = \hbar m \ket{j,m} \; \; \; J^2\ket{j,m} = \hbar^2 j(j+1)\ket{j,m}\]and assuming \(\ket{j,m}\) are normalised \[J_+ \ket{j,m} = \sqrt{j(j+1) - m(m+1)} \ket{j,m+1}\]\[J_- \ket{j,m} = \sqrt{j(j+1) - m(m-1)} \ket{j, m-1}\]Note \(J_+ \ket{j,j} = 0, J_-\ket{j, -j} = 0\)

Q: Definition. The Runge-Lenz operator \(R\) is \[[...]\]
A: R = \frac{1}{2\mu}(P \times L - L \times P) - k \frac{X}{|X|}
Back Extra: where \(\mu = \frac{m_p m_e}{m_p + m_e}\) and \(k = \frac{e^2}{4 \pi \epsilon_0}\)

Q: What are the operators and method for calculating Clebsch-Gordon coefficients?
A: For subsystems with \(j_1, j _2\), we know the maximally aligned state will be \[\ket{j,j} = \ket{j_1,j_1} \ket{j_2, j_2}\]Then we can apply the lowering operator \(J_- = J_{1-} + J_{2-}\) to the equation and we have \[J_{-}\ket{j,m} = \sqrt{j(j+1) - m(m-1)} \ket{j, m-1}\]Then when dropping down to \(\ket{j-1, j-1}\), we know it will be in the form \[\ket{j-1, j-1} = a\ket{j_1, j _1}\ket{j_2, j _2-1} + b\ket{j_1, j_1 - 1} \ket{j_2, j_2}\]and that it will be orthogonal to \(\ket{j,j-1}\). We can repeat this process to find everything.

Q: Which particles are bosons and which are fermions?
A: Integer spin - Bosons
Half Integer spin - Fermions
Bosons are symmetric under particle exchange, fermions are antisymmetric

Q: Definition. We define the interaction picture state \(\ket{\psi_I(t)}\) as \[[...]\]
A: \ket{\psi_I(t)} = U_0^{-1} \ket{\psi_S(t)}
Back Extra: the \(\ket{\psi_S(t)}\) would evolve in accordance with TDSE so in the absence of a perturbation we'd have \(\ket{\psi_I(t)} = U_0^{-1}U_0 \ket{\psi(0)} \ket{\psi(0)}\) so the state would in fact be independent of time.

Q: What is the time evolution of the interaction picture state?
A: \[i \hbar \frac{\partial}{\partial t} \ket{\psi_i(t)} = U_0^{-1}(t) \Delta (t)U_0(t)\ket{\psi_I(t)}\]

Q: What is Fermi's Golden Rule?
A: If our perturbation takes the form \[\Delta(t) = \begin{cases} 0 & t<0 \\ \Delta e^{-i\omega t} + \Delta^{\dagger} e^{+i\omega t} & t > 0 \end{cases}\]we get the following probabilities that the pertuburbation excites the system from \(\ket{m} \to \ket{k}\)
\[|a_k(t)|^2 \approx \begin{cases} \frac{4 |\bra{k}\Delta \ket{m}|^2}{\hbar^2 (\omega_{km} - \omega)^2} \sin^2 \left ( \frac{\omega_{km} - \omega}{2} t\right ) & \text{absorption} \\ \frac{4 |\bra{k}\Delta \ket{m}|^2}{\hbar^2 (\omega_{km} + \omega)^2} \sin^2 \left ( \frac{\omega_{km} + \omega}{2} t\right ) & \text{stimulated emission}\end{cases}\]where \(E_k > E_m\) corresponds to system absorbing energy from the perturbation whereas \(E_k < E_m\) is when the system is prompted to release energy by the perturbation.

Q: What is the density operator?
A: \[\rho = \sum_{\alpha} p_{\alpha} \ket{\psi_{\alpha}} \bra{\psi_{\alpha}}\]where \(p_{\alpha}\) are the probabilities that it is in a state \(\alpha\)

Q: What are the three properties of the density operator?
A: \[\rho^{\dagger} = \rho\]\[\text{tr}_{\mathcal{H}}(\rho) = 1\]\[\bra{\chi} \rho \ket{\chi} \ge 0 \; \; \forall \ket{\chi} \in \mathcal{H}\]

Q: Definition. The von Neumann entropy \(S(\rho)\) associated to a density operator is given by \[[...]\]
A: S(\rho) = -\text{tr}_{\mathcal{H} }(\rho \ln \rho)
Back Extra: Maximum entropy is attained iff \(\rho = \rho_{max} = \frac{1}{\dim \mathcal{H}}1_{\mathcal{H}}\)

Q: Definition. Any \(\ket{\psi} \in \mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B\) such that [...] is called a product state. Otherwise entangled.
A: \(\ket{\psi} = \ket{\phi} \otimes \ket{\chi}\) for \(\ket{\phi} \in \mathcal{H}_A\) and \(\ket{\chi} \in \mathcal{H}_B\)

Q: Definition. The reduced density operator for subsystem A \[[...]\]
A: \rho_A = \text{tr}_{\mathcal{H}_B}(\rho_{AB})
Back Extra: Is useful if we only observe A.

Q: Definition. Entanglement entropy quantifies entanglement between A and B and is given by \[[...]\]
A: S_A = - \text{tr}_{\mathcal{A} }(\rho_A \log \rho_A)

Q: What is the Pauli exclusion principle?
A: The fact that the state of identical fermions must be totally antisymmetric implies that no two fermions can be put in exactly the same state simultaneously.

Q: How is first-order perturbation theory modified in the case of degeneracy?
A: For degenerate states, we diagonalize the perturbation Hamiltonian \( \Delta H \) within the degenerate subspace. If \( V \) is the degenerate subspace, the matrix elements of \( \Delta H \) in \( V \) are: \[ H_{ij} = \langle \psi_i | \Delta H | \psi_j \rangle \] where \( |\psi_i\rangle \) and \( |\psi_j\rangle \) are basis states of \( V \). The eigenvalues of this matrix give the first-order energy corrections, and the eigenvectors give the corrected states

Q: What is the key difference between first-order perturbation theory with and without degeneracy?
A: Without Degeneracy: The first-order energy correction is simply the expectation value of the perturbation \[ E_n^{(1)} = \langle n | \Delta H | n \rangle \]With Degeneracy: The perturbation Hamiltonian \( \Delta H \) must be diagonalized within the degenerate subspace to determine the corrected energies and states.

Q: What is the formula for the second-order energy correction in non-degenerate perturbation theory, and how is it derived?
A: The second-order energy correction for a state \( |n\rangle \) is: \[ E_n^{(2)} = \sum_{m \neq n} \frac{|\langle m | \Delta H | n \rangle|^2}{E_n^{(0)} - E_m^{(0)}} \]1. Start with the perturbed Schrödinger equation: \[ H | E_n \rangle = E_n | E_n \rangle, \quad H = H_0 + \lambda \Delta H \] Expand the energy and states: \[ E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \cdots \] \[ | n \rangle = | n^{(0)} \rangle + \lambda | n^{(1)} \rangle + \cdots \] 2. Substitute into the Schrödinger equation and collect terms proportional to \( \lambda^2 \). 3. Solve for \( E_n^{(2)} \) using orthogonality of states and the fact that \( | n^{(1)} \rangle \) depends on the off-diagonal matrix elements \( \langle m | \Delta H | n \rangle \) for \( m \neq n \). The result expresses the energy correction as a sum over the contributions of all other states \( |m\rangle \), weighted by the matrix elements of \( \Delta H \) and the energy difference \( E_n^{(0)} - E_m^{(0)} \).

Q: What are the Pauli matrices, and how are they related to spin-\(\frac{1}{2}\) systems?
A: The Pauli matrices are the 2x2 matrices representing spin operators for a spin-\(\frac{1}{2}\) particle. They are defined as: \[ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \] For spin-\(\frac{1}{2}\) systems, the spin operators \( S_x, S_y, S_z \) are related to the Pauli matrices as: \[ S_i = \frac{\hbar}{2} \sigma_i, \quad \text{for } i = x, y, z. \] These matrices are crucial in describing quantum spin systems and satisfy the commutation relations: \[ [\sigma_x, \sigma_y] = 2i\sigma_z, \quad [\sigma_y, \sigma_z] = 2i\sigma_x, \quad [\sigma_z, \sigma_x] = 2i\sigma_y. \]

Q: What role does momentum play in quantum mechanics with respect to translations?
A: In quantum mechanics, the momentum operator \( \hat{p} \) acts as the generator of spatial translations. A translation by \( \vec{a} \) is implemented by the unitary operator:
\[T(\vec{a}) = e^{-i \vec{a} \cdot \hat{p} / \hbar}\]
where \( \hat{p} \) is the momentum operator.

Q: How is translation invariance connected to the conservation of momentum?
A: If the Hamiltonian \( \hat{H} \) commutes with the translation operator \( T(\vec{a}) \), i.e., \[ [\hat{H}, T(\vec{a})] = 0, \] then the system is invariant under translations. By Noether's theorem, this implies that momentum is conserved.

Q: How are rotations related to angular momentum in quantum mechanics?
A: Rotations in space are generated by the angular momentum operator \( \hat{L} \). A rotation by an angle \( \theta \) about an axis \( \hat{n} \) is represented by the unitary operator: \[ R(\theta, \hat{n}) = e^{-i \theta (\hat{n} \cdot \hat{L}) / \hbar}. \] This operator transforms the wavefunction under a rotation.

Q: What is a parity transformation, and how does it act in quantum mechanics?
A: Parity transformation corresponds to the inversion of spatial coordinates, \( \vec{r} \to -\vec{r} \). It is implemented by the parity operator \( \hat{P} \), which acts on the wavefunction as: \[ \hat{P} \psi(\vec{r}) = \psi(-\vec{r}). \] The parity of a system is classified as: - \( +1 \): Even (symmetric wavefunction) - \( -1 \): Odd (antisymmetric wavefunction).

Q: What is intrinsic parity, and why is it important in particle physics?
A: Intrinsic parity is a property of particles that determines their behavior under parity transformations. For example: - Scalar particles (e.g., the Higgs boson) have parity \( +1 \). - Pseudoscalar particles (e.g., pions) have parity \( -1 \). The total parity of a system depends on the intrinsic parity of particles and the symmetry of their spatial wavefunction.

Q: What is reflection symmetry, and how does it relate to parity?
A: Reflection symmetry is the invariance of a system under a reflection of spatial coordinates (e.g., \( x \to -x \)). Parity is a special case of reflection symmetry where all coordinates are inverted simultaneously.

Q: What is the first-order transition probability for a quantum system under a time-dependent perturbation?
A: The probability for a transition from an initial state \( |m\rangle \) to a final state \( |k\rangle \) is given by: \[ P_{m \to k}(t) = \left| a_k(t) \right|^2, \] where the amplitude \( a_k(t) \) to first order in perturbation theory is: \[ a_k(t) = -\frac{i}{\hbar} \int_{t_0}^t e^{i\omega_{km}t'} \langle k | \Delta H(t') | m \rangle \, dt', \] with \( \omega_{km} = (E_k - E_m) / \hbar \).

Q: What are the angular momentum selection rules for electric dipole transitions?
A: For electric dipole transitions: \[ \Delta \ell = \pm 1, \quad \Delta m = 0, \pm 1. \] These rules arise from the matrix element \( \langle k | \vec{r} | m \rangle \), which is non-zero only if the angular momentum quantum numbers obey these changes.

Q: What is the parity selection rule for electric dipole transitions?
A: The wavefunctions of the initial and final states must have opposite parity. This is because the electric dipole operator \( \vec{r} \) has odd parity, so: \[ \text{Initial Parity} \times \text{Final Parity} = -1. \]

Q: What is a qubit, and how is it represented in quantum mechanics?
A: A qubit is the fundamental unit of quantum information, represented as a superposition of two basis states \( |0\rangle \) and \( |1\rangle \): \[ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle, \] where \( \alpha, \beta \in \mathbb{C} \) and \( |\alpha|^2 + |\beta|^2 = 1 \). The Bloch sphere provides a geometric representation of the qubit's state.

Q: What is the no-cloning theorem in quantum mechanics?
A: The no-cloning theorem states that it is impossible to create an exact copy of an arbitrary unknown quantum state. Formally, there is no unitary operation \( U \) such that: \[ U(|\psi\rangle \otimes |e\rangle) = |\psi\rangle \otimes |\psi\rangle \] for all states \( |\psi\rangle \), where \( |e\rangle \) is an ancillary state.

Q: What is entanglement, and why is it significant in quantum mechanics?
A: Entanglement is a quantum phenomenon where the state of a composite system cannot be expressed as a tensor product of the states of its subsystems. For example, the singlet state: \[ |\psi\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle) \] is entangled. Measurement on one subsystem immediately determines the state of the other, regardless of distance.

Q: How do classical determinism and quantum probability differ?
A: Classical Determinism: The state of a system completely determines all future states (e.g., Newtonian mechanics).
Quantum Probability: The outcomes of measurements are probabilistic and described by the wavefunction \( |\psi\rangle \). The probabilities are given by: \[ P(\text{outcome}) = |\langle \text{outcome} | \psi \rangle|^2 \]

Q: What is the GHZ state, and how does it violate Bell's inequality?
A: The GHZ state for three qubits is: \[ |\psi_{\text{GHZ}}\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle). \] Measurements on the GHZ state exhibit correlations that violate Bell's inequality more starkly than the singlet state, showing the incompatibility of local hidden variable theories with quantum mechanics.