Introduction To Quantum Mechanics
Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
Question
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Chapter 4.4, Problem 4.35P

(a)

To determine

The probability that would get +2 at time t, if the component of spin angular momentum is measured along the x direction.

(a)

Expert Solution
Check Mark

Answer to Problem 4.35P

The probability that would get +2 at time t, if the component of spin angular momentum is measured along the x direction is P+(x)(t)=12[1+sinαcos(γB0t)].

Explanation of Solution

Write the expression to find χ, Equation 4.163

  χ=(cos(α/2)eiγB0t/2sin(α/2)eiγB0t/2)        (I)

Write the expression for χ+(x), Equation 4.151

    χ+(x)=12(11)        (II)

Write the expression for c+(x)

    c+(x)=χ+(x)χ        (III)

Write the expression to fine the probability along x direction

    P+(x)(t)=|c+(x)|2

Substitute equation (I), (II), and (III) in the above equation and solve for P+(x)(t)

P+(x)(t)=|c+(x)|2=|χ+(x)χ|2=|12(11)(cos(α/2)eiγB0t/2sin(α/2)eiγB0t/2)|2=12[cos(α/2)eiγB0t/2+sin(α/2)eiγB0t/2][cos(α/2)eiγB0t/2+sin(α/2)eiγB0t/2]

Solving further,

P+(x)(t)=12[cos2α2+sin2α2+sinα2cosα2(eiγB0t/2+eiγB0t/2)]=12[1+2sinα2cosα2cos(γB0t)]=12[1+sinαcos(γB0t)]

Conclusion:

Thus, the probability that would get +2 at time t, if the component of spin angular momentum is measured along the x direction is P+(x)(t)=12[1+sinαcos(γB0t)].

(b)

To determine

The probability that would get +2 at time t, if the component of spin angular momentum is measured along the y direction.

(b)

Expert Solution
Check Mark

Answer to Problem 4.35P

The probability that would get +2 at time t, if the component of spin angular momentum is measured along the y direction is P+(y)(t)=12[1sinαcos(γB0t)].

Explanation of Solution

Write the expression for c+(y)

    c+(y)=χ+(y)χ        (IV)

Write the expression to fine the probability along x direction

    P+(y)(t)=|c+(y)|2

Substitute equation (I) and (IV) in the above equation and solve for P+(y)(t)

P+(y)(t)=|c+(y)|2=|χ+(y)χ|2=|12(1i)(cos(α/2)eiγB0t/2sin(α/2)eiγB0t/2)|2=12[cos(α/2)eiγB0t/2+isin(α/2)eiγB0t/2][cos(α/2)eiγB0t/2isin(α/2)eiγB0t/2]

Solving further,

P+(y)(t)=12[cos2α2+sin2α2+isinα2cosα2(eiγB0t/2eiγB0t/2)]=12[12sinα2cosα2cos(γB0t)]=12[1sinαcos(γB0t)]

Conclusion:

Thus, the probability that would get +2 at time t, if the component of spin angular momentum is measured along the y direction is P+(y)(t)=12[1sinαcos(γB0t)].

(c)

To determine

The probability that would get +2 at time t, if the component of spin angular momentum is measured along the z direction.

(c)

Expert Solution
Check Mark

Answer to Problem 4.35P

The probability that would get +2 at time t, if the component of spin angular momentum is measured along the z direction is P+(z)(t)=cos2α2.

Explanation of Solution

Write the expression for χ+(z)

    χ+(z)=(10)        (V)

Write the expression for c+(z)

    c+(z)=χ+(z)χ        (VI)

Write the expression to fine the probability along z direction

    P+(z)(t)=|c+(z)|2

Substitute equation (I), (V), and (VI) in the above equation and solve for P+(z)(t)

P+(z)(t)=|c+(z)|2=|χ+(z)χ|2=|(10)(cos(α/2)eiγB0t/2sin(α/2)eiγB0t/2)|2=12[cos(α/2)eiγB0t/2][cos(α/2)eiγB0t/2]

Solving further,

P+(z)(t)=cos2α2

Conclusion:

Thus, the probability that would get +2 at time t, if the component of spin angular momentum is measured along the z direction is P+(z)(t)=cos2α2.

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