Questions and Answers of Introduction To Quantum Mechanics

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 to its transpose conjugate.
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 principal quantum numbers of the initial and final
Magnetic frustration. Consider three spin-1/2 particles arranged on the corners of a triangle and interacting via the Hamiltonianwhere J is a positive constant. This interaction favors opposite
Consider the Stark effect (Problem 7.45) for the n = 3 states of hydrogen. There are initially nine degenerate states, Ψ3ℓm (neglecting spin, as before), and we turn on an electric field in the z
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: V(x) = α|x|; (b) The quartic potential: V(x) =
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 origin). This time a is an adjustable parameter.
(a) Use the variational principle to prove that first-order non-degenerate perturbation theory always overestimates (or at any rate never underestimates) the ground state energy.(b) In view of (a),
Evaluate D and X (Equations 8.46 and 8.47). Check your answers against Equations 8.48 and 8.49. 0=0(800)|2|0}, D a (8.46)
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 electron in the bonding orbital (Equation 8.38) and
For the “half-harmonic oscillator” (Example 9.3), make a plot comparing the normalized WKB wave function for n = 3 to the exact solution. You’ll have to experiment to determine how wide to make
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 Example 2.3.Example 2.2Example 2.3 (H) = f (x, 0)* ĤY
(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. Describe how you would construct from it a completely
(a) Under parity, a “true” scalar operator does not change:whereas a pseudoscalar changes sign. Show therefore that [П̂,f̂] = 0 for a “true” scalar, whereas [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 discontinuity at x = 0. You may want to use the result
(a) Compute (x) , (p) , (x2) and (p2) , for the states Ψ0 (Equation 2.60) and Ψ1 (Equation 2.63), by explicit integration. Comment: In this and other problems involving the harmonic
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 the formcan be handled by “completing the
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 quadruplet and the two doublets, using the notation
In molecular and solid-state applications, one often uses a basis of orbitals aligned with the cartesian axes rather than the basis used throughout this chapter. For example, the orbitalsare a basis
Two noninteracting particles (of equal mass) share the same harmonic oscillator potential, one in the ground state and one in the first excited state.(a) Construct the wave function, Ψ(x1,x2),
Find the allowed energies of the half harmonic oscillator(This represents, for example, a spring that can be stretched, but not compressed.) V (x) = [(1/2) mw²x², x > 0, x < 0. 8
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 total angular momentum quantum numbers? Suggest a
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, construct the three-particle states (analogous to
In Section 5.1 we found that for noninteracting particles the wave function can be expressed as a product of single-particle states (Equation 5.9)—or, for identical particles, as a
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 particles of spin 1/2. Construct the four lowest-energy
(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 along r1, so thatThe θ2 integral is easy, but be
(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 your results against Table 5.1.(b) Figure out the
The ground state of lithium. Ignoring electron–electron repulsion, construct the ground state of lithium (Z = 3). Start with a spatial wave function, analogous to Equation 5.41, but remember that
(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 would this predict in the case of the excited
Discuss (qualitatively) the energy level scheme for helium if (a) Electrons were identical bosons, (b) If electrons were distinguishable particles (but with the same mass and charge).
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 free electron gas model, and use your result
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, and give your answer in electron volts.(b) What is
(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 form(Don’t bother to determine the normalization
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 E(q) as defined implicitly by Equation 5.71. On
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 treated as a Fermi gas (Section 5.3.1). Given a
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 Figure 5.5. This requires no new calculation, for
Show that most of the energies determined by Equation 5.71 are doubly degenerate. What are the exceptional cases?Equation 5.71 ma cos (qa) = cos (ka) + sin (ka). ħ²k (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 α/a = 1eV.
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 have a net magnetization (magnetic dipole moment
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 spin-down states will be different:There will be more
Suppose you have three particles, and three distinct one-particle states (Ψa(x), Ψb(x), and Ψc(x)) are available. How many different three particle states can be constructed (a) If they are
The Stoner criterion. The free-electron gas model (Section 5.3.1) ignores the Coulomb repulsion between electrons. Because of the exchange force (Section 5.1.2), Coulomb repulsion has a stronger
The harmonic chain consists of N equal masses arranged along a line and connected to their neighbors by identical springs:where xj is the displacement of the jth mass from its equilibrium position.
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). Assuming constant density, the radius R of such
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 the relativistic formula,Momentum is related to
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 rotation.(b) Show that, for Ψ expressed in polar
In this problem you will establish the correspondence between Equations 6.30 and 6.31.(a) Diagonalize the matrixto obtain the matrix M' = SMS-1 where S-1 is the unitary matrix whose columns
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 imagine the electrons to be confined to a box
Prove Equation 6.8. You may assume that Q (x̂,p̂) can be written in a power seriesfor some constants amn. 00 00 DO e(f, p) = EE annxm ри m=0 n=0
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 V(|x1-x2|) , so that the Hamiltonian isActing on a
Show that Equation 6.12 follows from Equation 6.11. (x-a)= e e-iga (x). (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 form of Equation 6.12.(a) Show that u satisfies 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 elements of V̂. As an example, Equation 6.52 with the
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. x = p' = ¹ * = -Î, == ¹ pĤ = -p, (6.18) (6.19)
Spin angular momentum, Ŝ, is even under parity, just like orbital angular momentum L̂:Acting on a spinor written in the standard basis (Equation 4.139), the parity operator becomes a 2 x 2 matrix.
Show how Equation 6.34 guarantees that a scalar is unchanged by a rotation: f̂ = R̂+ f̂ R̂ = f̂. [¹₁₁] = 0. (6.34)
(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 the matrix D in V̂' = D V̂. [2.₁] = (6.33)
Consider the matrix elements of L̂ between two definite-parity states: (n'ℓ'm'|L̂|nℓm). Under what conditions is this matrix element guaranteed to vanish? Note that the same selection rule
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 eigenstate of L̂2 and L̂z with eigenvalues ℓ(ℓ+1)
Consider the action of an infinitesimal rotation about the n axis of an angular momentum eigenstate Ψnℓm. Show thatand find the complex numbers Dm'm (they will depend on δ, n, and ℓ as well as
Consider the free particle in one dimension: Ĥ = p̂2/2m. This Hamiltonian has both translational symmetry and inversion symmetry.(a) Show that translations and inversion don’t commute.(b)
(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 you already derived one of these.(b) Derive
Show that the commutator [L̂,f̂] = 0 leads to the same rule, Equation 6.46, as does the commutator [L̂,f̂] = 0. (n'lm|ƒ\nlm) = (n'l (m + 1) |ƒ\n€ (m +1)). (6.46)
For an electron in the hydrogen statefind (r) after first expressing it in terms of a single reduced matrix element. 4 = 1 금 (V211 + 21-1),
Work out p̂H (t) for the system in Example 6.7 and comment on the correspondence with the classical equation of motion. Example 6.7 A particle of mass m moves in one dimension in a
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 the expectation value. V || 1 √2 (211 + 210)
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 between these equations and the classical
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 obtain the differential equations for x̂H and
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 solution to the time-dependent Schrödinger
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 operator by its spectral decomposition,rather than its
Rotations on spin states are given by an expression identical to Equation 6.32, with the spin angular momentum replacing the orbital angular momentum:In this problem we will consider rotations of a
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 in factso that, in addition to the complex
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 allowed energies. Explain why the energies are not
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 increases slightly: k→ (1+∈)k. (Perhaps we
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 V0 is a constant with the dimensions of energy,
Apply perturbation theory to the most general two-level system. The unperturbed Hamiltonian isand the perturbation iswith Vba = Vab, Vaa and Vbb real, so that H is hermitian. As in section 7.1.1, λ
(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 odd n.(b) Calculate the second-order correction
Consider a charged particle in the one-dimensional harmonic oscillator potential. Suppose we turn on a weak electric field (E), so that the potential energy is shifted by an amount H' = -qEx.(a) Show
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 nonzero terms in the sum will suffice.(b) Using the
Let the two “good” unperturbed states bewhere α± and β± are determined (up to normalization) by Equation 7.27 (or Equation 7.29). Show explicitly that(a)(b)(c) v² =α+²+B+₂
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 frictionlessly on a circular wire of circumference
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 first order in ϵ. Emn = (m + 2) + (x + 2)
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 corrections to the energy of the ground state and
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 claim as the “natural” generalization of the case
Consider a quantum system with just three linearly independent states. Suppose the Hamiltonian, in matrix form, iswhere V0 is a constant, and ϵ is some small number (∈ << 1).(a) Write down
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 (Equation 7.56), s = -2 (Equation 7.57), and s = -3
(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 constant from first principles (i.e., without recourse
Find the (lowest-order) relativistic correction to the energy levels of the one-dimensional harmonic oscillator. Use the technique of Problem 2.12. Problem 2.12 Find (x). (p). (x²). (p²), and (T),
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 parts, show thatCheck that the boundary term vanishes
Derive the fine structure formula (Equation 7.68) from the relativistic correction (Equation 7.58) and the spin-orbit coupling (Equation 7.67). E = (E₁₁)² 2mc² 4n + 1/2 3 (7.58)
Consider the (eight) n = 2 states, |2ℓjmj). Find the energy of each state, under weak-field Zeeman splitting, and construct a diagram like Figure 7.10 to show how the energies evolve as Bext
The exact fine-structure formula for hydrogen (obtained from the Dirac equation without recourse to perturbation theory) isExpand to order a4 (noting that a << 1), and show that you recover
Use Equation 7.60 to estimate the internal field in hydrogen, and characterize quantitatively a “strong” and “weak” Zeeman field. B = -L. 4л€ mc²r3 (7.60)

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