Modern Physics
Modern Physics
3rd Edition
ISBN: 9781111794378
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
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Chapter 15, Problem 22P

(a)

To determine

The expression for ντ(t).

(a)

Expert Solution
Check Mark

Answer to Problem 22P

The expression for ντ(t) is ντ(t)=sinθν2eiE2t/+cosθν3eiE3t/.

Explanation of Solution

Write the expression for ντ(t).

  ντ(t)=sinθν2eiE2t/+cosθν3eiE3t/        (I)

Here, ντ(t) the time evolution of ντ, θ is the mixing angle, E2 is the energy of the stationary state ν2 , E3 is the energy of the stationary state ν3 , t is the time and is the reduced Planck’s constant.

Conclusion:

Therefore, the expression for ντ(t) is ντ(t)=sinθν2eiE2t/+cosθν3eiE3t/.

(b)

To determine

The mathematical expression to show that the neutrino created at t=0 is entirely a μ neutrino.

(b)

Expert Solution
Check Mark

Answer to Problem 22P

The mathematical expressions to show that the neutrino created at t=0 is entirely a μ neutrino are (cosθν2sinθν3)*(cosθν2sinθν3)=1(sinθν2+cosθν3)*(sinθν2+cosθν3)=0 .

Explanation of Solution

Write the conditions for the neutrino created at t=0 to be entirely a μ neutrino.

  |vμ(0)|2=1        (II)

Here, |vμ(0)|2 is the probability for the particle to be found in νμ state at t=0 .

  |vτ(0)|2=0        (III)

Here, |vτ(0)|2 is the probability for the particle to be found in ντ state at t=0 .

Write the expression for νμ(t) .

  νμ(t)=cosθν2eiE2t/sinθν3eiE3t/        (IV)

Substitute 0 for t in equation (IV).

  νμ(0)=cosθν2e0sinθν3e0=cosθν2sinθν3        (V)

Put equation (V) in equation (II).

  (cosθν2sinθν3)*(cosθν2sinθν3)=1        (VI)

Substitute 0 for t in equation (I).

  ντ(0)=sinθν2e0+cosθν3e0=sinθν2+cosθν3        (VII)

Put equation (VII) in equation (III).

  (sinθν2+cosθν3)*(sinθν2+cosθν3)=0

Conclusion:

Therefore, the mathematical expressions to show that the neutrino created at t=0 is entirely a μ neutrino are (cosθν2sinθν3)*(cosθν2sinθν3)=1(sinθν2+cosθν3)*(sinθν2+cosθν3)=0 .

(c)

To determine

To show that the probability of finding a τ neutrino at time t is P(vμvτ)=sin22θ2(1cosE2E3t) .

(c)

Expert Solution
Check Mark

Answer to Problem 22P

It is showed that the probability of finding a τ neutrino at time t is P(vμvτ)=sin22θ2(1cosE2E3t) .

Explanation of Solution

Write the equation for the probability of finding a τ neutrino at time t .

  P(vμvτ)=|vτ(t)|2

Here, P(vμvτ) is the probability of finding a τ neutrino at time t .

Put equation (I) in the above equation.

  P(vμvτ)=(sinθν2eiE2t/+cosθν3eiE3t/)*(sinθν2eiE2t/+cosθν3eiE3t/)

After some algebra, it can be showed that

  P(vμvτ)=sin22θ2(1cosE2E3t)

Conclusion:

Therefore, it is showed that the probability of finding a τ neutrino at time t is P(vμvτ)=sin22θ2(1cosE2E3t) .

(d)

To determine

To show that for a fixed momentum p and small masses m2 and m3 , E2E3(m22m32)c32p .

(d)

Expert Solution
Check Mark

Answer to Problem 22P

It is showed that for a fixed momentum p and small masses m2 and m3 , E2E3(m22m32)c32p .

Explanation of Solution

In this case, the experiment involves setting up a μ-beam at t=0 with a definite momentum p .

Write the expression for E2 .

  E2=pc1+(m2c2)2(pc)2

Here, m2 is the mass of ν2 .

Rewrite the above equation for very small masses such that m2c2<<pc .

  E2pc(1+(m2c2)22(pc)2)        (VIII)

Write the expression for E3 .

  E3=pc1+(m3c2)2(pc)2

Here, m3 is the mass of ν3 .

Rewrite the above equation for very small masses such that m3c2<<pc .

  E3pc(1+(m3c2)22(pc)2)        (IX)

Find E2E3 using equations (VIII) and (IX).

  E2E3=pc(1+(m2c2)22(pc)2)pc(1+(m3c2)22(pc)2)=(m2c2)22(pc)2(m3c2)22(pc)2=(m22m32)c32p

Conclusion:

Therefore, it is showed that for a fixed momentum p and small masses m2 and m3 , E2E3(m22m32)c32p .

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