Exercise 5.8 Prove the following relation: [Îz, sin(29)] = 2iħ (sin² q - cos² ), where is the azimuthal angle. Hint: [A, BC] = B[Â, C] + [A, B] Ĉ. φ
Exercise 5.8 Prove the following relation: [Îz, sin(29)] = 2iħ (sin² q - cos² ), where is the azimuthal angle. Hint: [A, BC] = B[Â, C] + [A, B] Ĉ. φ
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Please help with EXERCISE 5.8 AND 5.13
![tators: (a) [α, 2] = Fħ±, (b) [Îz, R+] = ±ħR+, and (c) [Îz, 2] = 0.
Exercise 5.5
Prove the following two relations: R. Î
326
Exercise 5.7
Prove the following relation:
Exercise 5.6
The Hamiltonian due to the interaction of a particle of spin S with a magnetic field B is given
by Ĥ = -S. B where S is the spin. Calculate the commutator [S, Ĥ].
where is the azimuthal angle.
Exercise 5.8
Prove the following relation:
= 0 and P. Î = 0.
= 2iħ (sin² p - cos² ),
where is the azimuthal angle. Hint: [Â, ÂC] = Ê[Â, Ĉ] + [A, B] Ĉ.
[Îz, cos q] = iħ sin q,
[Îz, sin(20)]
Exercise 5.11
Consider the wave function
Exercise 5.9
Using the properties of Ĵ and Ĵ_, calculate |j, ±j) and [j, ±m) as functions of the action of
Ĵ on the states |j, ±m) and [j, ±j), respectively.
CHAPTER 5. ANGULAR MOMENTUM
Exercise 5.10
Consider the operator  = {(ĴxĴy + ĴyĴx).
(a) Calculate the expectation value of  and A² with respect to the state | j, m).
(b) Use the result of (a) to find an expression for A² in terms of: Ĵ¹, Ĵ², 3², Ĵ4, Ĵª.
y (0,0) = 3 sin 0 cose - 2(1 cos² )e²i.
(a) Write y (0, p) in terms of the spherical harmonics.
(b) Write the expression found in (a) in terms of the Cartesian coordinates.
(c) Is (0, p) an eigenstate of 1² or Îz?
(d) Find the probability of measuring 2ħ for the z-component of the orbital angular momen-
tum.
5.9. EXERCISES
Exercise 5.12
Show that ο(cos² — sin² y + 2i sin y cos p) = 2ħ²io, where is the azimuthal angle.
4
Exercise 5.13
Find the expressions for the spherical harmonics Y30(0, 9) and Y3,±1 (0, 0),
Y30 (0,0) = √7/167 (5 cos³ 0 - 3 cos 0),
in terms of the Cartesian coordinates x, y, z.
Exercise 5.14
(a) Show that the following expectation values between [1m) states satisfy the relations
(Îx) = (Îy) · = 0 and (β) = (Î}) = { [1(1 + 1)ħ² − m²ħ²].
(b) Verify the inequality ALxALy ≥ ħ² m/2, where ALx = √
Y3, +1 (0,0) = √21/64π sin 0(5 cos² 0-1) etio,
Exercise 5.16
Consider a system which is described by the state
√
8
√(L²) — (Lx)².
Exercise 5.15
A particle of mass m is fixed at one end a rigid rod of negligible mass and length R. The
other end of the rod rotates in the xy plane about a bearing located at the origin, whose axis is
in the z-direction.
y (0, 9) = Y₁1 (0, 0)+,
8
(a) Write the system's total energy in terms of its angular momentum L.
(b) Write down the time-independent Schrödinger equation of the system. Hint: In spherical
coordinates, only o varies.
(c) Solve for the possible energy levels of the system, in terms of m and the moment of
inertia Im R².
(d) Explain why there is no zero-point energy.
327
Y10(0, 9) + AY₁,-1(0, 0),](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F900ce081-76b5-4728-be49-0a5172963ecc%2F3058dd48-bd85-487a-bb3f-d504b7ba218f%2F8wwlvqs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:tators: (a) [α, 2] = Fħ±, (b) [Îz, R+] = ±ħR+, and (c) [Îz, 2] = 0.
Exercise 5.5
Prove the following two relations: R. Î
326
Exercise 5.7
Prove the following relation:
Exercise 5.6
The Hamiltonian due to the interaction of a particle of spin S with a magnetic field B is given
by Ĥ = -S. B where S is the spin. Calculate the commutator [S, Ĥ].
where is the azimuthal angle.
Exercise 5.8
Prove the following relation:
= 0 and P. Î = 0.
= 2iħ (sin² p - cos² ),
where is the azimuthal angle. Hint: [Â, ÂC] = Ê[Â, Ĉ] + [A, B] Ĉ.
[Îz, cos q] = iħ sin q,
[Îz, sin(20)]
Exercise 5.11
Consider the wave function
Exercise 5.9
Using the properties of Ĵ and Ĵ_, calculate |j, ±j) and [j, ±m) as functions of the action of
Ĵ on the states |j, ±m) and [j, ±j), respectively.
CHAPTER 5. ANGULAR MOMENTUM
Exercise 5.10
Consider the operator  = {(ĴxĴy + ĴyĴx).
(a) Calculate the expectation value of  and A² with respect to the state | j, m).
(b) Use the result of (a) to find an expression for A² in terms of: Ĵ¹, Ĵ², 3², Ĵ4, Ĵª.
y (0,0) = 3 sin 0 cose - 2(1 cos² )e²i.
(a) Write y (0, p) in terms of the spherical harmonics.
(b) Write the expression found in (a) in terms of the Cartesian coordinates.
(c) Is (0, p) an eigenstate of 1² or Îz?
(d) Find the probability of measuring 2ħ for the z-component of the orbital angular momen-
tum.
5.9. EXERCISES
Exercise 5.12
Show that ο(cos² — sin² y + 2i sin y cos p) = 2ħ²io, where is the azimuthal angle.
4
Exercise 5.13
Find the expressions for the spherical harmonics Y30(0, 9) and Y3,±1 (0, 0),
Y30 (0,0) = √7/167 (5 cos³ 0 - 3 cos 0),
in terms of the Cartesian coordinates x, y, z.
Exercise 5.14
(a) Show that the following expectation values between [1m) states satisfy the relations
(Îx) = (Îy) · = 0 and (β) = (Î}) = { [1(1 + 1)ħ² − m²ħ²].
(b) Verify the inequality ALxALy ≥ ħ² m/2, where ALx = √
Y3, +1 (0,0) = √21/64π sin 0(5 cos² 0-1) etio,
Exercise 5.16
Consider a system which is described by the state
√
8
√(L²) — (Lx)².
Exercise 5.15
A particle of mass m is fixed at one end a rigid rod of negligible mass and length R. The
other end of the rod rotates in the xy plane about a bearing located at the origin, whose axis is
in the z-direction.
y (0, 9) = Y₁1 (0, 0)+,
8
(a) Write the system's total energy in terms of its angular momentum L.
(b) Write down the time-independent Schrödinger equation of the system. Hint: In spherical
coordinates, only o varies.
(c) Solve for the possible energy levels of the system, in terms of m and the moment of
inertia Im R².
(d) Explain why there is no zero-point energy.
327
Y10(0, 9) + AY₁,-1(0, 0),
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