Problem 18: Scattering at a hard sphere 2+3+1 = 6 points We study the scattering of particles at a hard sphere with radius a.

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Friedrich-Schiller-Universität Jena Winter term 2017/18 Prof. Dr. Andreas Wipf

Dr. Luca Zambelli

Problems in Advanced Quantum Mechanics Problem Sheet 8

Problem 18: Scattering at a hard sphere 2+3+1 = 6 points We study the scattering of particles at a hard sphere with radius a.

1. Determine the total scattering cross section σ = P

` σ _` for the elastic scattering of particles at a hard sphere with radius a (the de Broglie wave length satisfies λ a).

2. Determine the dimensionless ratio σ/(2πa ² ) with your favorite computer program (mat- lab, octave, mathematica,..) for ka = 1, 2, 3, . . . , 50 and plot the result.

3. Compare the result for fast particles (ka >> 1) with the classical cross section.

Hint: You may probably need

i tan δ _` = e ^2iδ

^`

− 1

e ^2iδ

^`

+ 1 and sin ² δ _` = tan ² δ _` 1 + tan ² δ _`

Problem 19: Gaussian integral 3 points

Let A be a real and symmetric matrix. Prove the formula

Z ^N Y

i=1

dx i e

²ⁱ^~

^x

^T

^Ax = (2iπ ~ ) ^N/2 1

√ det A .

Hint: transform first to coordinates for which A is diagonal.

Problem 20: Action of one-dimensional harmonic oscillator 4+1 = 5 points Show that the action along the classical path from (t ₁ , x ₁ ) to (t ₂ , x ₂ ) is

S[x ₂ , t ₂ ; x ₁ , t ₁ ] = mω

2 sin(ωT ) (x ² ₁ + x ² ₂ ) cos(ωT ) − 2x ₁ x ₂

, T = t ₂ − t ₁ ,

where ω is the circular frequency of the oscillator. The propagator is given by

K (x ₁ , t ₂ ; x ₁ ; t ₁ ) = mω 2πi ~ sin ωT

1/2

e ^iS(x

²

^,t

²

^;x

¹

^,t

¹

^)/~

Show that this propagator fulfills lim _T _→0 K (x ₂ , t ₂ ; x ₁ , t ₁ ) = δ(x ₂ − x ₁ ).

Hint: Concerning the last question, recall what is the propagator for a free particle.

Submission date: Thursday, 15. December 2017, before the lecture begins.

Problem 18: Scattering at a hard sphere 2+3+1 = 6 points We study the scattering of particles at a hard sphere with radius a.

Friedrich-Schiller-Universität Jena Winter term 2017/18 Prof. Dr. Andreas Wipf

Dr. Luca Zambelli

Problems in Advanced Quantum Mechanics Problem Sheet 8

Problem 18: Scattering at a hard sphere 2+3+1 = 6 points We study the scattering of particles at a hard sphere with radius a.

1. Determine the total scattering cross section σ = P

` σ ` for the elastic scattering of particles at a hard sphere with radius a (the de Broglie wave length satisfies λ a).

2. Determine the dimensionless ratio σ/(2πa 2 ) with your favorite computer program (mat- lab, octave, mathematica,..) for ka = 1, 2, 3, . . . , 50 and plot the result.

3. Compare the result for fast particles (ka >> 1) with the classical cross section.

Hint: You may probably need

i tan δ ` = e 2iδ

− 1

e 2iδ

+ 1 and sin 2 δ ` = tan 2 δ ` 1 + tan 2 δ `

Problem 19: Gaussian integral 3 points

Let A be a real and symmetric matrix. Prove the formula

Z N Y

i=1

dx i e

x

Ax = (2iπ ~ ) N/2 1

√ det A .

Hint: transform first to coordinates for which A is diagonal.

Problem 20: Action of one-dimensional harmonic oscillator 4+1 = 5 points Show that the action along the classical path from (t 1 , x 1 ) to (t 2 , x 2 ) is

S[x 2 , t 2 ; x 1 , t 1 ] = mω

2 sin(ωT ) (x 2 1 + x 2 2 ) cos(ωT ) − 2x 1 x 2

, T = t 2 − t 1 ,

where ω is the circular frequency of the oscillator. The propagator is given by

K (x 1 , t 2 ; x 1 ; t 1 ) = mω 2πi ~ sin ωT

1/2

e iS(x

,t

;x

,t

)/~

Show that this propagator fulfills lim T →0 K (x 2 , t 2 ; x 1 , t 1 ) = δ(x 2 − x 1 ).

Hint: Concerning the last question, recall what is the propagator for a free particle.

Submission date: Thursday, 15. December 2017, before the lecture begins.

` σ _` for the elastic scattering of particles at a hard sphere with radius a (the de Broglie wave length satisfies λ a).

2. Determine the dimensionless ratio σ/(2πa ² ) with your favorite computer program (mat- lab, octave, mathematica,..) for ka = 1, 2, 3, . . . , 50 and plot the result.

i tan δ _` = e ^2iδ

e ^2iδ

+ 1 and sin ² δ _` = tan ² δ _` 1 + tan ² δ _`

Z ^N Y

^x

^Ax = (2iπ ~ ) ^N/2 1

Problem 20: Action of one-dimensional harmonic oscillator 4+1 = 5 points Show that the action along the classical path from (t ₁ , x ₁ ) to (t ₂ , x ₂ ) is

S[x ₂ , t ₂ ; x ₁ , t ₁ ] = mω

2 sin(ωT ) (x ² ₁ + x ² ₂ ) cos(ωT ) − 2x ₁ x ₂

, T = t ₂ − t ₁ ,

K (x ₁ , t ₂ ; x ₁ ; t ₁ ) = mω 2πi ~ sin ωT

e ^iS(x

^,t

^;x

^,t

^)/~

Show that this propagator fulfills lim _T _→0 K (x ₂ , t ₂ ; x ₁ , t ₁ ) = δ(x ₂ − x ₁ ).