- Prove that the components of a vector with respect to a given basis are unique.
- Calculate the lifetime (in seconds) for each of the four n = 2 states of hydrogen.
- Show that if an operator is hermitian, then its matrix elements in any orthonormal basis satisfy That is, the corresponding matrix is equal
- Calculate (zĤz), in the ground state of hydrogen.
- Coincident spectral lines. According to the Rydberg formula (Equation 4.93) the wavelength of a line in the hydrogen spectrum is determined by the
- Magnetic frustration. Consider three spin-1/2 particles arranged on the corners of a triangle and interacting via the Hamiltonianwhere J is a
- Consider the Stark effect (Problem 7.45) for the n = 3 states of hydrogen. There are initially nine degenerate states, Ψ3ℓm (neglecting spin, as
- Use a gaussian trial function (Equation 8.2) to obtain the lowest upper bound you can on the ground state energy of (a) The linear potential:
- Find the best bound on for the delta function potential V(x) = -δ(x), using a triangular trial function (Equation 8.10, only centered at the
- (a) Use the variational principle to prove that first-order non-degenerate perturbation theory always overestimates (or at any rate never
- Evaluate D and X (Equations 8.46 and 8.47). Check your answers against Equations 8.48 and 8.49.
- Show that the antisymmetric state (Equation 8.56) can be expressed in terms of the molecular orbitals of Section 8.3—specifically, by placing one
- For the “half-harmonic oscillator” (Example 9.3), make a plot comparing the normalized WKB wave function for n = 3 to the exact solution.
- For the wave function in Example 2.2, find the expectation value of H, at time t = 0, the “old fashioned” way:Compare the result we got in
- (a) Suppose you could find a solution Ψ (r1,r2, ..., rz) to the Schrödinger equation (Equation 5.37), for the Hamiltonian in Equation 5.36.
- (a) Under parity, a “true” scalar operator does not change:whereas a pseudoscalar changes sign. Show therefore that [П̂,f̂] = 0 for a
- Check the uncertainty principle for the wave function in Equation 2.132. Calculating (p2) can be tricky, because the derivative of Ψ has a step
- (a) Compute (x) , (p) , (x2) and (p2) , for the states Ψ0 (Equation 2.60) and Ψ1 (Equation 2.63), by explicit integration. Comment: In
- A free particle has the initial wave functionwhere A and a are (real and positive) constants.(a) Normalize Ψ (x,0).(b) Find Ψ (x,t). Integrals of
- If you combine three spin- particles, you can get a total spin of 3/2 or 1/2 (and the latter can be achieved in two distinct ways). Construct the
- In molecular and solid-state applications, one often uses a basis of orbitals aligned with the cartesian axes rather than the basis used throughout
- Two noninteracting particles (of equal mass) share the same harmonic oscillator potential, one in the ground state and one in the first excited
- Find the allowed energies of the half harmonic oscillator(This represents, for example, a spring that can be stretched, but not compressed.)
- The ground state of dysprosium (element 66, in the 6th row of the Periodic Table) is listed as 5I8. What are the total spin, total orbital, and grand
- Suppose you had three particles, one in state Ψa, one in state Ψb(x), and one in state Ψc(x). Assuming Ψa, Ψb, and Ψc are orthonormal,
- In Section 5.1 we found that for noninteracting particles the wave function can be expressed as a product of single-particle states (Equation
- In Example 5.1 and Problem 5.5(b) we ignored spin (or, if you prefer, we assumed the particles are in the same spin state).(a) Do it now for
- (a) Calculate ((1/|r1 - r2|)) for the state Ψ0 (Equation 5.41). Do the d3r2 integral first, using spherical coordinates, and setting the polar axis
- (a) Figure out the electron configurations (in the notation of Equation 5.44) for the first two rows of the Periodic Table (up to neon), and check
- The ground state of lithium. Ignoring electron–electron repulsion, construct the ground state of lithium (Z = 3). Start with a spatial wave
- (a) Hund’s first rule says that, consistent with the Pauli principle, the state with the highest total spin (S) will have the lowest energy. What
- Discuss (qualitatively) the energy level scheme for helium if (a) Electrons were identical bosons, (b) If electrons were distinguishable
- The bulk modulus of a substance is the ratio of a small decrease in pressure to the resulting fractional increase in volume:Show that B = (5/3) P, in
- The density of copper is 8.96 g/cm3, and its atomic weight is 63.5 g/mole.(a) Calculate the Fermi energy for copper (Equation 5.54). Assume d = 1,
- (a) Using Equations 5.66 and 5.70, show that the wave function for a particle in the periodic delta function potential can be written in the
- Find the average energy per free electron (Etot/Nd), as a fraction of the Fermi energy.
- Make a plot of E vs. q for the band structure in Section 5.3.2. Use α = 1 (in units where m = ћ = a = 1). In Mathematica, ContourPlot will graph
- Helium-3 is fermion with spin 1/2 (unlike the more common isotope helium-4 which is a boson). At low temperatures (T << TF) , helium-3 can be
- Suppose we use delta function wells, instead of spikes (i.e. switch the sign of α in Equation 5.64). Analyze this case, constructing the analog to
- Show that most of the energies determined by Equation 5.71 are doubly degenerate. What are the exceptional cases?Equation 5.71
- Find the energy at the bottom of the first allowed band, for the case β = 10, correct to three significant digits. For the sake of argument, assume
- Consider a free electron gas (Section 5.3.1) with unequal numbers of spin-up and spin-down particles (N+ and N- and respectively). Such a gas would
- Pauli paramagnetism. If the free electron gas (Section 5.3.1) is placed in a uniform magnetic field B = Bk̂, the energies of the spin-up and
- Suppose you have three particles, and three distinct one-particle states (Ψa(x), Ψb(x), and Ψc(x)) are available. How many different three
- The Stoner criterion. The free-electron gas model (Section 5.3.1) ignores the Coulomb repulsion between electrons. Because of the exchange force
- The harmonic chain consists of N equal masses arranged along a line and connected to their neighbors by identical springs:where xj is the
- Calculate the Fermi energy for electrons in a two-dimensional infinite square well. Let σ be the number of free electrons per unit area.
- Certain cold stars (called white dwarfs) are stabilized against gravitational collapse by the degeneracy pressure of their electrons (Equation 5.57).
- We can extend the theory of a free electron gas (Section 5.3.1) to the relativistic domain by replacing the classical kinetic energy, E = P2/2m, with
- Consider the parity operator in three dimensions.(a) Show that ^∏Ψ(r) = Ψ'(r) = Ψ(-r) is equivalent to a mirror reflection followed by a
- In this problem you will establish the correspondence between Equations 6.30 and 6.31.(a) Diagonalize the matrixto obtain the matrix M' =
- In Section 5.3.1 we put the electrons in a box with impenetrable walls. The same results can be obtained using periodic boundary conditions. We still
- Prove Equation 6.8. You may assume that Q (x̂,p̂) can be written in a power seriesfor some constants amn.
- Consider two particles of mass m1 and m2 (in one dimension) that interact via a potential that depends only on the distance between the particles
- Show that Equation 6.12 follows from Equation 6.11.
- Consider a particle of mass m moving in a potential V(x) with period a. We know from Bloch’s theorem that the wave function can be written in the
- Show that, for a Hermitian operator Q̂, the operator Û = exp [-iQ̂] is unitary.
- Show that the operator p̂' obtained by applying a translation to the operator p̂ is p̂' = T̂+ p̂T̂ = p̂.
- (a) Sandwich each of the six commutation relations in Equations 6.52–6.54 between (n'ℓ'm'| and |nℓm) to obtain relations between matrix
- Show that the position and momentum operators are odd under parity. That is, prove Equations 6.18, 6.19, and, by extension, 6.21 and 6.22.
- Spin angular momentum, Ŝ, is even under parity, just like orbital angular momentum L̂:Acting on a spinor written in the standard basis (Equation
- Show how Equation 6.34 guarantees that a scalar is unchanged by a rotation: f̂ = R̂+ f̂ R̂ = f̂.
- (a) Show that the parity operator П̂̂ is Hermitian.(b) Show that the eigenvalues of the parity operator are ±1.
- Working from Equation 6.33, find how the vector operator V̂ transforms for an infinitesimal rotation by an angle δ about the y axis. That is, find
- Consider the matrix elements of L̂ between two definite-parity states: (n'ℓ'm'|L̂|nℓm). Under what conditions is this matrix element guaranteed
- For any vector operator V̂ one can define raising and lowering operators as(a) Using Equation 6.33, show that(b) Show that, if Ψ is an
- Consider the action of an infinitesimal rotation about the n axis of an angular momentum eigenstate Ψnℓm. Show thatand find the complex numbers
- Consider the free particle in one dimension: Ĥ = p̂2/2m. This Hamiltonian has both translational symmetry and inversion symmetry.(a) Show that
- (a) Show that the commutation relations, Equations 6.50 6.54, follow from the definition of a vector operator, Equation 6.33. If you did Problem 6.19
- Show that the commutator [L̂,f̂] = 0 leads to the same rule, Equation 6.46, as does the commutator [L̂,f̂] = 0.
- For an electron in the hydrogen statefind (r) after first expressing it in terms of a single reduced matrix element.
- Work out p̂H (t) for the system in Example 6.7 and comment on the correspondence with the classical equation of motion.
- Express the expectation value of the dipole moment Pe for an electron in the hydrogen statein terms of a single reduced matrix element, and evaluate
- Consider a free particle of mass m. Show that the position and momentum operators in the Heisenberg picture are given by Comment on the relationship
- Differentiate Equation 6.72 to obtain the Heisenberg equations of motion(for Q̂ and Ĥ independent of time). Plug in Q̂ = x̂ and Q̂ = p̂ to
- Consider a time-independent Hamiltonian for a particle moving in one dimension that has stationary states Ψn(x) with energies En.(a) Show that the
- In deriving Equation 6.3 we assumed that our function had a Taylor series. The result holds more generally if we define the exponential of an
- Rotations on spin states are given by an expression identical to Equation 6.32, with the spin angular momentum replacing the orbital angular
- As an angular momentum, a particle’s spin must flip under time reversal (Problem 6.36). The action of time-reversal on a spinor (Section 4.4.1) is
- Suppose we put a delta-function bump in the center of the infinite square well:where α is a constant.(a) Find the first-order correction to the
- For the harmonic oscillator [V(x) = (1/2)kx2], the allowed energies arewhere ω = √k/m is the classical frequency. Now suppose the spring constant
- Two identical spin-zero bosons are placed in an infinite square well (Equation 2.22). They interact weakly with one another, via the potential(where
- Apply perturbation theory to the most general two-level system. The unperturbed Hamiltonian isand the perturbation iswith Vba = Vab, Vaa and Vbb
- (a) Find the second-order correction to the energies (E2n) for the potential in Problem 7.1. Comment: You can sum the series explicitly, obtainingfor
- Consider a charged particle in the one-dimensional harmonic oscillator potential. Suppose we turn on a weak electric field (E), so that the potential
- Consider a particle in the potential shown in Figure 7.3.(a) Find the first-order correction to the ground-state wave function. The first three
- Let the two “good” unperturbed states bewhere α± and β± are determined (up to normalization) by Equation 7.27 (or Equation 7.29). Show
- Consider a particle of mass m that is free to move in a one-dimensional region of length L that closes on itself (for instance, a bead that slides
- Show that the first-order energy corrections computed in Example 7.3 (Equation 7.34) agree with an expansion of the exact solution (Equation 7.21) to
- Suppose we perturb the infinite cubical well (Problem 4.2) by putting a delta function “bump” at the point (a/4,a/2,3a/4):Find the first-order
- In the text I asserted that the first-order corrections to an n-fold degenerate energy are the eigenvalues of the W matrix, and I justified this
- Consider a quantum system with just three linearly independent states. Suppose the Hamiltonian, in matrix form, iswhere V0 is a constant, and ϵ is
- In Problem 4.52 you calculated the expectation value of rs in the state Ψ321. Check your answer for the special cases s = 0 (trivial), s = -1
- (a) Express the Bohr energies in terms of the fine structure constant and the rest energy (mc2) of the electron.(b) Calculate the fine structure
- Find the (lowest-order) relativistic correction to the energy levels of the one-dimensional harmonic oscillator. Use the technique of Problem 2.12.
- Show that P2 is hermitian, for hydrogen states with ℓ = 0.For such states is independent of θ and ϕ, so(Equation 4.13). Using integration by
- Derive the fine structure formula (Equation 7.68) from the relativistic correction (Equation 7.58) and the spin-orbit coupling (Equation 7.67).
- Consider the (eight) n = 2 states, |2ℓjmj). Find the energy of each state, under weak-field Zeeman splitting, and construct a diagram like Figure
- The exact fine-structure formula for hydrogen (obtained from the Dirac equation without recourse to perturbation theory) isExpand to order a4 (noting
- Use Equation 7.60 to estimate the internal field in hydrogen, and characterize quantitatively a “strong” and “weak” Zeeman field.