Concept explainers
(a)
The
(a)
Answer to Problem 11.29P
The
Explanation of Solution
Write the expression to find the Hamiltonian matric for a spinning charge particle in a magnetic field
Here,
Write the expression for appropriate spin matrix
Where,
Given, the total magnetic field
Where,
Solving equation (I),
Substitute equation (II) in the above equation,
Solving further,
Conclusion:
Thus, the
(b)
Show that
(b)
Answer to Problem 11.29P
It has been proved that
Explanation of Solution
Write the time-dependent Schrodinger equation
Given,
Thus,
Substitute the above relations in equation (IV)
From part (a),
Thus,
Hence proved.
Conclusion:
It has been proved that
(c)
Whether the general solution for
(c)
Answer to Problem 11.29P
It has been proved that the general solution for
Explanation of Solution
From part (b),
Using the above relations,
Differentiate
Rewrite the above equation in the form of second order differential equation,
Solving the terms inside the bracket,
Substitute the above relation in equation (V)
The characteristic equation is
Solving the above quadratic equation,
Where,
The general solution
The general solution
For the initial condition,
From part (b),
Substitute
Substitute equation (IX) and (X) in (VII)
And
Hence proved.
Conclusion:
It has been proved that the general solution for
(d)
The probability of a transition to spin down, as a function of time.
(d)
Answer to Problem 11.29P
The probability of a transition to spin down, as a function of time is
Explanation of Solution
Write the expression to find the probability of transition to spin down
Given,
Substitute
Now, the probability of a transition to spin down is
Conclusion:
Thus, the probability of a transition to spin down, as a function of time is
(e)
Sketch the resonance curve,
(e)
Answer to Problem 11.29P
The resonance curve,
the full width at half maximum is
Explanation of Solution
Given,
Taking square root on both sides,
Thus, the full width at half maximum,
Conclusion:
The resonance curve,
the full width at half maximum is
(f)
The resonant frequency in a nuclear magnetic resonance experiment and the width of the resonance curve.
(f)
Answer to Problem 11.29P
The resonant frequency in a nuclear magnetic resonance experiment is
Explanation of Solution
Compare Equation 4.156 and 7.89, the gyro-magnetic ratio for a proton is
Where,
Write the expression to find the resonant frequency
Where,
Substitute equation (XI) in the above equation,
Substitute
Write the expression to find the width of the resonance curve
Where,
Thus, the above equation becomes,
Substitute
Conclusion:
Thus, the resonant frequency in a nuclear magnetic resonance experiment is
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Chapter 11 Solutions
Introduction To Quantum Mechanics
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