Physics Problems & Solutions: GR0877

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Quantum Mechanics - Commutation Relations

Let $\hat{\mathbf{J}}$ be a quantum mechanical angular momentum operator. The commutator [ $\hat{J}_x$ $\hat{J}_y$ , $\hat{J}_x$ ] is equivalent to which of the following?

A. 0
B. iħ $\hat{J}_z$
C. iħ $\hat{J}_z$ $\hat{J}_x$
D. −iħ $\hat{J}_x$ $\hat{J}_z$
E. iħ $\hat{J}_x$ $\hat{J}_y$

(GR0877 #95)

Solution:

Commutator identities: [A, B] = − [B, A]

and, [B, AC] = A[B, C] + [B, A]C

[ $\hat{J}_x$ $\hat{J}_y$ , $\hat{J}_x$ ] = − [ $\hat{J}_x$ , $\hat{J}_x$ $\hat{J}_y$ ]

A = $\hat{J}_x$
B = $\hat{J}_x$
C = $\hat{J}_y$

[ $\hat{J}_x$ , $\hat{J}_x$ $\hat{J}_y$ ] = $\hat{J}_x$ [ $\hat{J}_x$ , $\hat{J}_y$ ] + [ $\hat{J}_x$ , $\hat{J}_x$ ] $\hat{J}_y$
= $\hat{J}_x$ [ $\hat{J}_x$ , $\hat{J}_y$ ] + 0
= $\hat{J}_x$ [iħ $\hat{J}_z$ ] + 0

− [ $\hat{J}_x$ , $\hat{J}_x$ $\hat{J}_y$ ] = −iħ $\hat{J}_x$ $\hat{J}_z$

Answer: D

Nuclear & Particle Physics - Compton Scattering

In the Compton effect, a photon with energy E scatters through 90^o angle from stationary electron mass m. The energy of the scattered photon is

A. E
B. E/2
C. E²/mc²
D. E²/(E + mc²)
E. E∙mc²/(E + mc²)

(GR0877 #97)

Solution:

Compton scattering: λ_f − λ_i = ^h/_mc(1− cosθ)
Photon energy, E = ^hc/_λ

^hc/_{E_f} − ^hc/_E = ^h/_mc(1 − cos 90^o)
¹/_{E_f} − ¹/_E = ¹/_mc_²(1 − 0)
¹/_{E_f}  = ¹/_m_c_²+ ¹/_E
^Em^c²/_{E_f}  = E + mc²
^Em^c²/_{E_f}  = E + mc²
E_f = ^Em^c²/_{E +}_mc_²

Answer: E

Nuclear & Particle Physics - Muon Decay

Which of the following is the principal decay mode of the positive muon μ⁺?

A. μ⁺ → e⁺ + ν_e
B. μ⁺ → p + ν_μ
C. μ⁺ → n + e⁺ + ν_e
D. μ⁺ → e⁺ + ν_e+ v̄_μ
E. μ⁺ → π⁺+ v̄_e+ ν_μ

(GR0877 #98)

Solution:

(A) FALSE
μ⁺ → e⁺ + ν_eviolates lepton number: 1 → 1 + 1

(B), (C) FALSE
Muon is elementary particle (lepton), p and n are composite particle (bosons). Elementary particle cannot decay into composite particle, (B), (C) FALSE

(D) TRUE

(E) FALSE
π is pion, which a boson, not a lepton.

Answer: D

Classical Mechanics - Rotational Motion

A small particle of mass m is at rest on a horizontal circular platform that is free to rotate about a vertical axis through its center. The particle is located at a radius r from the axis, as shown in the figure above. The platform begins to rotate with constant angular acceleration α. Because of friction between the particle and the platform, the particle remains at rest with respect to the platform. When the platform has reached angular speed ω, the angle θ between the static frictional for f_sand the inward radial direction is given by which of the following?

A. θ = ω²r/g
B. θ = ω²/α
C. θ = α/ω²
D. θ = tan⁻¹(ω²/α)
E. θ = tan⁻¹(α/ω²)

(GR0877 #99)

Solution:

f sin θ = Iα = mrα
f cos θ = mω²r

tan θ = α/ω²

Answer: E