Quantum Mechanics

Question 1

Define the Heisenberg Uncertainty Principle and explain its significance in quantum mechanics.

Answer 1

• In the context of position and momentum, the principle states that it is impossible to simultaneously know the exact position and momentum of a particle.
• Mathematically, it is expressed as:
  Δx Δp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck constant.
• For two arbitrary observables A and B, the uncertainty principle can be written in terms of their commutator, namely, ΔA ΔB ≥ ½ |⟨[A, B]⟩|.

It implies inherent limitations in measurement, challenging classical determinism and underscoring the probabilistic nature of quantum mechanics.

Question 2

What are the key postulates of quantum mechanics?

Answer 2

• The state of a quantum system is described by a vector that lives in a Hilbert space.
• Observables are represented by Hermitian operators, with measurement results given by the eigenvalues and the post-measurement states by the eigenstates.
• If the state is |ψ⟩, then the probability of measuring the eigenvalue aₙ of observable A is p(aₙ) = |⟨aₙ|ψ⟩|². This is Born’s postulate.
• The evolution of a quantum state is governed by the time-dependent Schrödinger equation.

These postulates introduce superposition, probabilistic interpretation of measurement, and time evolution of the quantum state.

Question 3

True or False:

The superposition principle allows a quantum system to exist in multiple states simultaneously until a measurement is made.

Answer 3

True

The superposition principle states that a system can exist in a linear combination of eigenstates until measurement collapses the state.

Superposition arises because quantum states live in a Hilbert space.

Question 4

Explain the concept of wave-particle duality.

Answer 4

Wave-particle duality posits that particles such as electrons exhibit both wave-like and particle-like properties.

The double-slit experiment demonstrates interference patterns typical of waves, yet individual particle impacts are observed on the detector.

Whether we see the wave nature or the particle nature depends on what we are looking for.

Question 5

What is the time-independent Schrödinger equation for a particle moving in one-dimensional potential V(x)?

Answer 5

• ħ is the reduced Planck constant
• m is the particle’s mass
• ψ_n(x) is the nth energy eigenfunction (in position space)
• E_n is the corresponding energy

Note that this is the energy eigenvalue equation written in position space.

Question 6

Identify the differences between fermions and bosons within the context of quantum mechanics.

Answer 6

• Fermions: Particles that follow Fermi-Dirac statistics and have half-integer spins. They obey the Pauli exclusion principle. The state of two fermions must be antisymmetric under particle exchange.
• Bosons: Particles that follow Bose-Einstein statistics and have integer spins. They can occupy the same quantum state. The state of two bosons must be symmetric under particle exchange.

Question 7

What is the significance of commutation relations in quantum mechanics?

Answer 7

• Commutation relations determine whether two observables can be simultaneously measured with arbitrary precision.
• If two observables commute, [A, B] = 0, they share a common set of eigenstates and can be simultaneously measured.

The commutator of an observable A with the Hamiltonian H also governs the rate of change of the expectation value of A. Symmetries are also studied via commutation relations. We have a symmetry if the unitary transformation corresponding to the symmetry commutes with the Hamiltonian.

Question 8

True or False:

Consider three operators A, B, and C.

If A commutes with B, and B commutes with C, then A commutes with C.

Answer 8

False

For example, consider the angular momentum operators, A = L_z, B = L², and C = L_x. Then [A,B] = [B,C] = 0, but [A,C] is not zero.

We are not dealing with a transitive property here.

Question 9

Fill in the blank:

In quantum mechanics, the probability density associated with a wave function (ψ(x)) is given by the equation ______.

Answer 9

The probability density ( |ψ(x)|² ) gives the probability of finding the particle, whose wavefunction is ψ(x)) between x and x + dx as |ψ(x)|² dx.

Question 10

Analyze the role of potential energy barriers in quantum tunneling.

Answer 10

Quantum tunneling allows particles to pass through potential energy barriers higher than their kinetic energy.

It arises from the wave-like nature of particles. A particle will bounce off such a barrier, but a wave can penetrate the barrier.

This permits non-zero probability amplitudes on the other side of the barrier.

Question 11

Explain the concept of quantum entanglement.

Answer 11

Entanglement is a quantum phenomenon where the quantum states of two or more particles become interdependent, regardless of the distance between them.

This happens when the state of the two particles cannot be written as the tensor product of states of the individual particles.

Measurements on one particle instantly affect the state of the other, challenging classical notions of locality and separability.

Question 12

True or False:

The expectation value of an observable in quantum mechanics is always one of its eigenvalues.

Answer 12

False

The expectation value is a weighted average of all possible measurement outcomes (eigenvalues), based on their probabilities in a given quantum state. It is not necessarily equal to any single eigenvalue unless the system is in a definite eigenstate.

Only if the system is in an eigenstate of the operator A will the expectation value equal the corresponding eigenvalue.

Otherwise, the result is a weighted average of multiple eigenvalues and is generally not itself an eigenvalue. If the system is

Question 13

What is the Born rule and how does it relate to quantum measurements?

Answer 13

The Born rule states that the probability of obtaining a particular measurement outcome is equal to the absolute value squared of the amplitude of the corresponding eigenstate.

It fundamentally connects the mathematical formalism of quantum mechanics with experimental observations. Remember that the measurement outcomes are given by the eigenvalues.

Question 14

Explain the concept of a Hilbert space in quantum mechanics.

Answer 14

A Hilbert space is a complete vector space with an inner product, providing the framework for quantum states.

It allows for the representation and manipulation of wave functions, facilitating the application of linear algebra in quantum mechanics.

Superposition, which is an experimental fact, necessitates the use of a vector space. Using a ‘complete’ vector space ensures that there are no ‘holes’ in the vector space. Finally, the presence of the inner product ensures that we can calculate probabilities via the Born rule.

Question 15

What happens to a quantum state when we perform an ideal projective measurement?

Answer 15

The measurement collapses the state to an eigenstate of the measured observable, the eigenstate corresponding to the measurement result obtained (the eigenvalue).

This makes sense because if we perform a second measurement of the same observable right after the right measurement, with probability one we should get the same measurement result.

Question 16

Fill in the blanks:

The eigenvalues of the Hamiltonian operator correspond to the ______ ______ of the quantum system.

Answer 16

energy levels

The Hamiltonian in quantum mechanics has a dual role. The Hamiltonian operator governs the time evolution of a quantum system, and its eigenvalues represent the quantized energy levels accessible to the system.

Question 17

What is the time-independent Schrödinger equation for a particle in a one-dimensional infinite potential well?

Explain the boundary conditions that need to be imposed.

Answer 17

• Consider a potential V(x) defined as 0 for 0 < x < a and infinity elsewhere. The particle can then not be found outside the region 0 < x < a, so we only need to solve the Schrodinger equation in this region.
• The time-independent Schrödinger equation is -ħ²/2m d²ψ(x)/dx² = Eψ(x). This has sinusoidal solutions.
• The boundary conditions that need to be imposed are ψ(0) = ψ(a) = 0. This ensures that the wavefunction is continuous, since the wavefunction (and hence the probability density) outside the region 0 < x < a is zero.

The boundary conditions arise from the requirement that the wavefunction must vanish at the walls of the infinite potential well (which follows from the requirement that the wavefunction is continuous).

Question 18

Why are the energy levels of a particle in a one-dimensional infinite potential well quantized?

Answer 18

The wavefunction must be zero at the boundaries of the well, so only standing wave solutions with specific wavelengths are allowed.

This leads to discrete (quantized) values of momentum and energy.

The boundary conditions force the wavefunction to be sinusoidal with nodes at the walls.

Only integer multiples of half-wavelengths fit exactly in the well, restricting the allowed energies.

Question 19

True or False:

Consider the bound state of a particle in a one-dimensional potential well. This bound state cannot be degenerate.

Answer 19

True

This follows from the mathematical structure of the one-dimensional time-independent Schrodinger equation (as well as the boundary condition that the wavefunction needs to vanish infinitely far away), which is an ordinary differential equation.

The same conclusion does not hold in two or three dimensions, since we are then dealing with a partial differential equation.

Question 20

True or False:

In the harmonic oscillator potential, the spacing between adjacent energy levels is constant.

Answer 20

True

n is a non-negative integer, which leads to constant spacing ħω.

Question 21

What is the significance of the zero-point energy in the quantum harmonic oscillator?

Answer 21

• The ground state energy is called the zero-point energy.
• It demonstrates that even in the lowest energy state, the oscillator possesses non-zero energy.

Zero-point energy arises due to the Heisenberg uncertainty principle. If the kinetic energy was zero, we would precisely know the position of the particle.

Therefore, the average kinetic energy would not be zero. That means that since the particle is moving in a potential well, the average potential energy would also not be zero.

Question 22

Fill in the blank:

The probability density for finding a particle in the ground state of a one-dimensional quantum harmonic oscillator is highest at the _____ or minimum of the potential.

Answer 22

center

The ground state wavefunction ψ₀(x) is a Gaussian, centered at the potential minimum (which is at the center of the potential, typically chosen to be x = 0), leading to the highest probability density at the center.

Question 23

What are the creation and annihilation (raising and lowering) operators for the 1D quantum harmonic oscillator?

Answer 23

Question 24

For the quantum harmonic oscillator, what is [N,a] and [N,a^†] where N = a^†a is the number operator?

Answer 24

[N,a] = -a
[N,a^†] = a^†

• Use [AB,C] = A[B,C] + [A,C]B
• Use [a,a^†] = 1

These relations are essential in interpreting the operators a and a^†
and lowering and raising operators.

Question 25

# True or False: For the **quantum harmonic oscillator**, we can find eigenstates of the raising operator a^†.

Answer 25

False ## Footnote If such an eigenstate |ψ⟩ exists, a^† |ψ⟩ = λ|ψ⟩. We can write |ψ⟩ as a linear combination of the number states. Using this in the eigenvalue equation, we find that the left hand side will not contain the ground state |0⟩, while the right hand side will. Therefore, the two sides cannot be equal. The lowering (or annihilation) operator does have eigenstates. These states are called coherent states and play a very important role in quantum optics.

Question 26

Describe the concept of **degeneracy in the context of the hydrogen** atom energy levels.

Answer 26

* It refers to **multiple quantum states having the same energy**. * For hydrogen, energy levels depend only on the principal quantum number n, leading to degeneracy in l, m_l and m_s quantum numbers. ## Footnote The degree of degeneracy for a given n is (2n^2), accounting for all combinations of l, m_l and m_s that produce the same energy.

Question 27

What potential energy is used in the **Schrödinger equation for hydrogenic atoms**?

Answer 27

The potential energy is assumed to be **Coulombic**. ## Footnote This simplifies the problem to a central force problem, allowing separation of variables in spherical coordinates. Note that the solutions obtained ignore relativistic effects.

Question 28

Explain the physical significance of the **quantum number (l) in the hydrogen atom.**

Answer 28

Knowing l for a state means that the state has a **well-defined total orbital angular momentum**. ## Footnote l also tells us something about which spherical harmonic goes into writing the state. In this way, (l) determines the shape of the electron's probability distribution. For example, l = 0 states have a spherical probability distribution. l takes on values l = 0, 1, 2, 3, ..., n - 1. (l) influences the angular part of the wavefunction but not directly the energy levels in the hydrogen atom due to degeneracy. This degeneracy is due to the 1/r potential. In alkali atoms, the effective potential that the valence electron sees is not proportional to 1/r due to shielding effects - the energy levels then do depend on l.

Question 29

# True or False: In a spherically symmetric potential, the wavefunction can be expressed as a product of **radial and angular components.**

Answer 29

True ## Footnote In such potentials, the Schrödinger equation allows separation of variables into radial and spherical harmonics components, simplifying the solution process. This can also be seen as a consequence of rotational symmetry.

Question 30

# Fill in the blanks: The angular momentum of a particle in a **central potential** is conserved because the potential is \_\_\_\_\_\_ \_\_\_\_\_\_.

Answer 30

spherically symmetric ## Footnote This implies that the system has rotational invariance, leading to conservation of angular momentum.

Question 31

What is the significance of the **normalization condition** for wavefunctions in quantum mechanics?

Answer 31

Ensures the total **probability of finding the particle over all space is 1.** ## Footnote The normalization condition is crucial for the probabilistic interpretation of quantum mechanics. This is what allows the interpretation of |ψ(x)|² as a probability density.

Question 32

Describe the role of **boundary conditions in solving the Schrödinger equation** for the bound states in a one-dimensional **finite potential well.**

Answer 32

* The Schrodinger equation needs to be solved in three regions now: to the left of the well, within the well, and to the right of the well. * The solutions need to satisfy the requirement that the wavefunction does not blow up when infinitely far away from the well. * The wavefunctions also need to be continuous at the walls of the potential well. Moreover, the first derivative of the wavefunctions also needs to be continuous. * These requirements lead to a discrete set of energy levels with the corresponding discrete set of energy eigenfunctions. ## Footnote For a finite well, the energy eigenfunctions are sinusoidal inside the well and exponentially decaying outside. They are continuous and their first derivative is also continuous.

Question 33

State the form of the **potential energy in a 3D isotropic quantum harmonic oscillator**. What are the energy levels?

Answer 33

## Footnote The isotropic three-dimensional harmonic oscillator can be thought of as being composed of three independent one-dimensional harmonic oscillators. Note that the energy levels are degenerate. There are many different n_x, n_y, and n_z that give the same n, but due to the different n_x, n_y, and n_z, the quantum state is different. Degeneracy is expected here due to the rotational symmetry. If H is the Hamiltonian, and we have [H, A] = [H, B] = 0 but [A, B] does not equal 0, then we will have degeneracy. Here, A = L_z and B = L_x satisfy these requirements.

Question 34

Explain how the **radial wavefunction of the hydrogen atom changes** with increasing principal quantum number (n).

Answer 34

The radial wavefunction becomes more **spread out**, with the average radial distance increasing with n. ## Footnote This is expected since increasing n increases the energy, and corresponds to what we expect from the Bohr model. The number of nodes that we can have also increases, since the number of nodes for the radial wavefunction is given by n - l - 1 (the number of angular nodes is l, so the total number of nodes is n - 1). Higher (n) leads to larger orbits, reflecting the increased average potential energy and radial expectation values.

Question 35

What is the physical interpretation of the **wavefunction's phase** in quantum mechanics?

Answer 35

* The **overall global phase does not affect measurable quantities directly**. This phase disappears when calculating measurement probabilities. * However, relative phases can impact interference and superposition states. ## Footnote In particular, consider a two-level system with basis states |0⟩ and |1⟩. Take two quantum states to be |ψ₁⟩ = (1/√2) (|0⟩ + |1⟩) and |ψ₂⟩ = 1/√2 (|0⟩ - |1⟩). These two states are physically distinguishable by measuring the observable σₓ = |0⟩⟨1| + |1⟩⟨0| - |ψ₁⟩ is an eigenstate of this operator (with eigenvalue +1) while |ψ₂⟩ is an eigenstate of this operator with eigenvalue -1. While the **absolute phase is not observable**, the relative phase is crucial in quantum interference phenomena.

Question 36

# True or False: The bound state **energy levels** of a particle in a finite potential well are always less than the potential height.

Answer 36

True ## Footnote Due to the finite potential barrier, bound states have energy levels below the potential height, allowing the particle to remain confined. If the energy of the particle is greater than the potential height, then the particle can escape (we are then dealing with scattering states).

Question 37

Describe how the concept of **parity** is used to classify the **energy eigenfunctions** in quantum mechanics.

Answer 37

If the Hamiltonian of a system is invariant under parity (that is, inversion), then **the energy eigenfunctions will also be eigenfunctions of the parity operator**. They will have eigenvalues (for the parity operator) +1 (even) or -1 (odd). ## Footnote For example, the usual Hamiltonian for hydrogen is invariant under parity. The energy eigenfunctions then have well-defined parity. In fact, the energy eigenstate |n, l, mₗ⟩ has parity (−1)ˡ. For one-dimensional problems, if the potential V(x) is an even function, then the energy eigenfunctions will be either even or odd.

Question 38

A particle is in a state with **total angular momentum l=2**. What are the possible outcomes of a measurement of L_z?

Answer 38

## Footnote m_ℓ goes in integer steps from -ℓ all the way to +ℓ

Question 39

Explain how the **Stern-Gerlach experiment** demonstrates the quantization of spin angular momentum.

Answer 39

Silver atoms passing through a non-uniform magnetic field **split into two distinct paths**, not a continuous spread. This shows their **spin has only two possible values**, proving that spin is quantized. ## Footnote The discrete deflection is due to the intrinsic angular momentum (spin) having only specific allowed orientations.

Question 40

What are the **commutation relations for angular momentum operators**?

Answer 40

## Footnote These can be derived by considering angular momentum as the generator of rotations, and then using the fact that the order of rotations matters (this is true even in classical physics). One of the implications of the commutation relations is that components of angular momentum cannot be simultaneously known with arbitrary precision.

Question 41

# True or False: The **spin quantum** number for an electron is an integer.

Answer 41

False ## Footnote The spin quantum number for an electron is s = 1/2.

Question 42

# Fill in the blank: The **eigenvalues** of the S_z operator for a spin-(1/2) particle are \_\_\_\_\_\_.

Answer 42

## Footnote These values correspond to the two possible spin states, "spin up" and "spin down," along the z-axis.

Question 43

What is the significance of the **Clebsch-Gordan coefficients** in the context of angular momentum?

Answer 43

They **relate the uncoupled basis** (individual angular momenta) **to the coupled basis** (total angular momentum) when combining two angular momenta. ## Footnote They are essential for calculating transition amplitudes in atomic systems. For example, they are used with the Wigner-Eckart theorem.

Question 44

Outline the differences between **orbital and spin angular momentum.**

Answer 44

* **Orbital Angular Momentum**: Arises from particle motion around a point. It is quantized by non-negative integer values of \( l \). We can write down the orbital angular momentum eigenstates in position space (these are the spherical harmonics). * **Spin Angular Momentum**: Intrinsic property of particles. It is quantized by non-negative half-integer or integer values of \( s \). Spin lives in its own Hilbert space. ## Footnote Orbital momentum depends on position and momentum; spin is an internal quantum property present even at rest.

Question 45

How does the expectation value of the spin angular momentum ⟨**S**⟩ evolve in the presence of a constant magnetic field? ## Footnote Assume zero orbital angular momentum.

Answer 45

The expectation value of the spin angular momentum ⟨**S**⟩ precesses around the magnetic field. ## Footnote The magnetic moment is proportional to the spin angular momentum, and the rate of change of the angular momentum is the torque, which is the cross product of the magnetic moment and the magnetic field. Precession affects phenomena like nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR).

Question 46

What is the **magnetic moment associated with the spin of an electron**?

Answer 46

## Footnote gₛ is the g-factor, approximately 2 for electrons The electron is a charged particle that carries angular momentum. It is expected, therefore, that there is an associated magnetic moment.

Question 47

Analyze the relationship between **angular momentum and rotational symmetry** in physical systems.

Answer 47

Rotational symmetry implies **conservation of angular momentum.** ## Footnote Noether's theorem directly relates symmetries to conservation laws.

Question 48

Two particles each have spin-1/2. What are the possible **total spin quantum numbers**, and how many states correspond to each?

Answer 48

Total spin: * S=1 (triplet, 3 states) * S=0 (singlet, 1 state) ## Footnote When it comes to adding angular momenta j₁ and j₂, the rule is that the possible values of the total angular momentum quantum number are j₁ + j₂, j₁ +j₂ - 1..., |j₁ -j₂|. For each j, the corresponding values of m_j are from -j, -j + 1, ..., j.

Question 49

A spin-1/2 particle is in quantum state |0⟩, an eigenstate of σ_z with eigenvalue +1. If we measure the operator −i|0⟩⟨1| + i|1⟩⟨0|, **what are the possible outcomes and their probabilities?**

Answer 49

* The operator is σ_y, which has eigenvalues +1 and -1. * The probabilities of measuring each are both 1/2. ## Footnote Probabilities are found using the Born rule and the fact that |0 is an equal superposition of the σ_y eigenstates.

Question 50

Why must the wave functions of **identical fermions** be **antisymmetric** under particle exchange?

Answer 50

Due to the **symmetrization postulate**: identical fermions must have antisymmetric wavefunctions, which leads to the Pauli exclusion principle. **No two fermions can share the same quantum state.** ## Footnote This behavior is tied to the spin-statistics theorem: particles with half-integer spin are fermions and obey Fermi-Dirac statistics.

Question 51

# True or False: The antisymmetry requirement for fermionic wave functions violates the **superposition principle** of quantum mechanics.

Answer 51

False ## Footnote The superposition principle still holds. The physical state of a system of identical fermions must be antisymmetric. We can still form superpositions of such antisymmetric states - the superposition state will also be antisymmetric.

Question 52

# Fill in the blank: The **wave functions** of identical bosons must be \_\_\_\_\_\_\_\_ under particle exchange.

Answer 52

symmetric ## Footnote This follows from the symmetrization postulate. This exchange symmetry is what allows bosons to occupy the same quantum state, leading to the phenomenon of Bose-Einstein condensation under appropriate conditions.

Question 53

What is the state of two identical bosons if one of them is in the state |ϕ⟩ and the other is in state |χ⟩, assuming that |ϕ⟩ is orthogonal to |χ⟩?

Answer 53

## Footnote This form ensures symmetry under the exchange of particles and satisfies the indistinguishability condition of quantum particles.

Question 54

What is the fundamental difference in the **statistical distribution of identical fermions and bosons?**

Answer 54

* Fermions follow **Fermi-Dirac statistics**, characterized by the Fermi-Dirac distribution. * Bosons follow **Bose-Einstein statistics**, characterized by the Bose-Einstein distribution. ## Footnote The difference arises from the difference under particle exchange. This seemingly small difference has profound implications in fields like solid-state physics and quantum optics.

Question 55

How does the symmetrization postulate relate to the **exchange interaction**?

Answer 55

* Two fermions must have an antisymmetric total state, so the symmetry of their spin state affects their spatial separation. * This leads to the exchange interaction, **where interaction energy depends on spin configuration**, even without direct spin-spin forces. ## Footnote The exchange interaction is a non-classical effect that cannot be explained by electrostatic interactions alone.

Question 56

Explain the role of **permutation operators** in determining the symmetry of a multi-particle wave function.

Answer 56

* They **exchange the quantum states** of individual particles in a quantum many-particle state. * For fermions, the application of any two-particle permutation operator introduces a negative sign, indicating an antisymmetric quantum state. * For bosons, the quantum state remains unchanged under such two-particle permutations, indicating a symmetric quantum state. ## Footnote These operators can help in formulating the correct wave functions for systems of identical particles.

Question 57

Explain how **time-independent perturbation theory** is applied to a non-degenerate quantum system.

Answer 57

* Start with an unperturbed Hamiltonian ( H₀) with known eigenstates and eigenvalues: 𝐻₀ |𝐸ₙ⁽⁰⁾⟩ = 𝐸ₙ⁽⁰⁾ |𝐸ₙ⁽⁰⁾⟩ * Introduce a small perturbation ( H' ) such that the total Hamiltonian is ( H = H₀ + λH' ), where λ is a small parameter. * Write the eigenvalues and eigenstates of the total Hamiltonian as power series in λ: |𝐸ₙ⟩ = |𝐸ₙ⁽⁰⁾⟩ + λ |𝐸ₙ⁽¹⁾⟩ + λ² |𝐸ₙ⁽²⁾⟩ + ⋯ 𝐸ₙ = 𝐸ₙ⁽⁰⁾ + λ𝐸ₙ⁽¹⁾ + λ²𝐸ₙ⁽²⁾ + ⋯ * Plug these in H|𝐸ₙ⟩ = 𝐸ₙ |𝐸ₙ⟩. * Match like powers of λ * Then take inner product with ⟨𝐸ₙ⁽⁰⁾|and simplify. ## Footnote For the correction to the state, expand the state in terms of the unperturbed eigenstates and find the expansion coefficients. λ is a dimensionless parameter introduced to control the strength of the perturbation. It is typically set to one at the end of calculations, but is kept symbolically small during expansions to track the order of approximation.

Question 58

In a **degenerate perturbation scenario**, how does one determine the correct set of states to use for **calculating first-order energy corrections**?

Answer 58

One diagonalizes the perturbation Hamiltonian within the degenerate subspace of the unperturbed Hamiltonian. The eigenstates of this matrix form the correct basis for calculating first-order energy shifts (they are the 'good states'). The eigenvalues are the energy shifts. ## Footnote This avoids divergences and ensures proper splitting of degenerate levels. One can often find the correct basis by symmetry considerations (rather than first constructing the matrix and diagonalizing it. On this basis, the perturbation Hamiltonian within the degenerate subspace is diagonal, simplifying the calculation of energy shifts.

Question 59

What is the **first-order correction** to the energy for a non-degenerate state in time-independent perturbation theory?

Answer 59

The first-order energy correction is given by the **expectation value** of the perturbation Hamiltonian in the corresponding unperturbed state. ## Footnote This expression can also be used to calculate the first-order energy correction even when there is degeneracy - the catch is that we must be using the 'good states'.

Question 60

What **assumption** must hold for the validity of using time-independent perturbation theory?

Answer 60

The perturbation Hamiltonian H' must be **small** compared to the unperturbed Hamiltonian H₀. ## Footnote What this means is that the perturbation causes only a small amount of 'mixing' between the unperturbed eigenstates. Mathematically, this means |⟨m|𝐻′|n⟩| ≪ |𝐸ₙ⁽⁰⁾ − 𝐸ₘ⁽⁰⁾|. Here, |n⟩ and |m⟩ are eigenstates of the unperturbed Hamiltonian with eigenvalues 𝐸ₙ⁽⁰⁾ and 𝐸ₘ⁽⁰⁾. This condition is clear if one looks at the first-order correction to the energy eigenstates. If the perturbation is weak, this correction should be small.

Question 61

What is the expression for the **second-order energy correction** in non-degenerate perturbation theory?

Answer 61

## Footnote This follows from power series expansions for the energy eigenstates and the energy eigenvalues.

Question 62

For a time-dependent Hamiltonian, why is finding the **time-evolution of a quantum state** generally a very hard problem?

Answer 62

The Hamiltonian at different times **does not commute** and is an operator. ## Footnote This means that finding the unitary time-evolution operator is not simple. Only in a few special cases can this be found analytically. We generally have to resort to either using time-dependent perturbation theory, or using numerical methods.

Question 63

A two-level system starts at |ψ(0)⟩ = 1/√2 (|0⟩ + |1⟩), with Hamiltonian H = (ħω ₀/2)σ_z. **What is the expectation value of σₓ at time t?**

Answer 63

⟨σₓ⟩(t) =cos(ω₀t). ## Footnote The time-evolved state is obtained by acting the time-evolution operator on the initial state.

Question 64

How does **time-dependent perturbation theory** help calculate transition probabilities between quantum states?

Answer 64

## Footnote * ∣ϕ _n⟩ are eigenstates of H_{0 * Using the Schrödinger equation and assuming a weak perturbation H^', solve for time-dependent coefficient cn(t). * The squared magnitude gives the transition probability to state ∣ϕ n⟩. This can also be derived using the Dyson series, a perturbative expansion of the time-evolution operator.}

Question 65

# Fill in the blank: In the context of time evolution in quantum mechanics, the term 'adiabatic' refers to changes that are \_\_\_\_\_ compared to the system's natural timescales.

Answer 65

slow ## Footnote The adiabatic theorem says that if we have a time-dependent Hamiltonian H(t), and our initial state is in the nth eigenstate of H(0), then if the Hamiltonian is changed slowly, at time t the state of our system is the nth eigenstate of H(t).

Question 66

How does the **variational principle** help us to find approximately the ground state energy of a Hamiltonian we cannot diagonalize exactly?

Answer 66

* Given a Hamiltonian, the variational principle states that the energy of the ground state E₀ satisfies the inequality ⟨ψ|H|ψ⟩ ≥ E₀, for any state |ψ⟩. * We choose a trial wavefunction with adjustable parameters, then compute ⟨ψ|H|ψ⟩ and minimize it. * This minimum gives an upper bound on the true ground state energy. ## Footnote The better the trial wavefunction approximates the true ground state, the more accurate the energy estimate.

Question 67

Consider two spin-1/2 particles in the singlet state. You measure the spin of the first particle to be ħ/2 in the z direction, and then you measure the spin of the second particle in the x direction. **What are the measurement results and their corresponding probabilities?**

Answer 67

* After the measurement of the first particle, the state of the second spin is spin down along z. * The measurement results then for this second spin are ħ/2 and -ħ/2, both with probability 1/2.

Question 68

# Fill in the blanks: In the context of **blackbody radiation**, the quantization of energy was first proposed by \_\_\_\_\_\_\_\_ \_\_\_\_\_\_ to explain the ultraviolet catastrophe.

Answer 68

Max Planck ## Footnote Planck introduced the concept of **quantized energy levels** to reconcile theoretical predictions with experimental observations of blackbody radiation.

Question 69

Discuss the significance of the **zero-point energy** in quantum systems.

Answer 69

* Zero-point energy is the **lowest possible energy** that a quantum mechanical system may have, and it can be non-zero due to the boundary conditions imposed on the energy eigenfunctions. * It implies that quantum systems exhibit **intrinsic fluctuations** even at absolute zero temperature. ## Footnote Zero-point energy is a consequence of the Heisenberg uncertainty principle, preventing a system from being completely at rest.