Introduction to Quantum Mechanics – Solution Manual 3rd. Edition by David Griths, Darrell Schroeter Reed College | Holooly

Introduction to Quantum Mechanics – Solution Manual [EXP-27105]

491 SOLVED PROBLEMS

Question: 4.40

(a) Aparticle of spin 1 and a particle of spin 2 are at rest in a configuration such that the total spin is 3, and its z component is h . If you measured the z-component of the angular momentum of the spin-2 particle, what values might you get, and what is the probability of each one? Comment: Using

Verified Answer:

(a) From the 2 × 1 Clebsch-Gordan table we get [la...

Question: 4.37

(a) Apply S- to |10〉 (Equation 4.175), and confirm that you get √2h |1-1〉. (b) Apply S± to |100〉 (Equation 4.176), and confirm that you get zero. (c) Show that | 11 〉 and |1-1〉 (Equation 4.175) are eigenstates of S^2 , with the appropriate eigenvalue.

Verified Answer:

(a)

S_{-}|10\rangle=\left(S_{-}^{(1)}+S_{-...

Question: 5.15

(a) Calculate 〈 ( 1/|r1 -r2| ) 〉 for the state ψ0 (Equation 5.41). Hint: Do the d^3 r2 integral first, using spherical coordinates, and setting the polar axis along r1, so that | r1 – r2 | = √r1^2 + r^2^2 – 2r1 r2 cos θ2 . The θ2 integral is easy, but be careful to take the positive root. You’ll

Verified Answer:

(a)

\left\langle\frac{1}{\left|r_{1}-r_{2}...

Question: 4.10

(a) Check that A rj1(kr) satisfies the radial equation with V(r)=0 and l = 1 . (b) Determine graphically the allowed energies for the infinite spherical well, when l = 1. Show that for large N, EN1 ≈ (h^2 π^2/2ma^2) (N+1/2)^2 Hint: First show that j1(x) = 0 ⇒ x = tan x . Plot x and tan x on the same

Verified Answer:

(a)

u=\operatorname{Arj}_{1}(k r)=A\left[\...

Question: 3.8

(a) Check that the eigenvalues of the hermitian operator in Example 3.1 are real. Show that the eigenfunctions (for distinct eigenvalues) are orthogonal. (b) Do the same for the operator in Problem 3.6.

Verified Answer:

(a) The eigenvalues (Eq. 3.29) are 0 , ±1, ±2 ,......

Question: 3.9

(a) Cite a Hamiltonian from Chapter 2 (other than the harmonic oscillator) that has only a discrete spectrum. (b) Cite a Hamiltonian from Chapter 2 (other than the free particle) that has only a continuous spectrum. (c) Cite a Hamiltonian from Chapter 2 (other than the finite square well) that has

Verified Answer:

(a) Infinite square well (Eq. 2.22).

V(x)=...

Question: 2.11

(a) Compute (x) , (p) , (x^2), and (p^2), for the states ψ0 (Equation 2.60) and ψ1 (Equation 2.63), by explicit integration. Comment: In this and other problems involving the harmonic oscillator it simplifies matters if you introduce the variable ξ ≡ √mω/hx and the constant α ≡ (mω/πh)^1/4. (b)

Verified Answer:

(a) Note that

ψ_0

is even, and [l...

Question: 12.7

(a) Construct the density matrix for an electron that is either in the state spin up along x (with probability 1/3) or in the state spin down along y (with probability 2/3). (b) Find 〈Sy〉 for the electron in (a).

Verified Answer:

(a) From Example 12.1, the density matrix for an e...

Question: 4.52

(a) Construct the spatial wave function (ψ) for hydrogen in the state n = 3 , l = 2 , m = 1. Express your answer as a function of r, θ, ϕ, and a (the Bohr radius) only—no other variables (ρ, z, etc.) or functions (Y, v, etc.), or constants (A, c0, etc.), or derivatives, allowed (π is okay, and e,

Verified Answer:

(a) From Tables 4.3 and 4.7, Table 4.3: The fi...

Question: 4.53

(a) Construct the wave function for hydrogen in the state n =4 , l = 3 , m = 3 . Express your answer as a function of the spherical coordinates r, θ, and ϕ. (b) Find the expectation value of r in this state. (As always, look up any nontrivial integrals.) (c) If you could somehow measure the

Verified Answer:

(a) From Tables 4.3 and 4.7, Table 4.3: The fi...

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