(a)
The normalized wave function.
(a)
Answer to Problem 2.42P
The normalized wave function is
Explanation of Solution
Given that the wave function is;
Normalize the wave function.
Use the normalization constant to rewrite the wave function.
Conclusion:
Therefore, the normalized wave function is
(b)
The wave function at time
(b)
Answer to Problem 2.42P
The wave function at time
Explanation of Solution
Write the solution to the generic quantum problem, for a free particle;
Use equation (III) in (IV).
Thus, the wave function at time
Let
The integral in equation (VII) becomes;
Thus, the wave function at time
In the above expression, the first exponential term which represent the Gaussian envelop travels at speed
Conclusion:
Therefore, the wave function at time
(c)
The probability density
(c)
Answer to Problem 2.42P
The probability density is
Explanation of Solution
The wave function is obtained from part (b) is given in equation (VIII).
The probability density can be expressed as;
Simplify the term in the square bracket.
The term
Where,
Thus, equation (IX) can be modified as;
Where
Thus, the graph of
Conclusion:
Therefore, the probability density is
(d)
The expectation values
(d)
Answer to Problem 2.42P
The expectation values and
Explanation of Solution
Write the expression for the expectation value of
Use equation (XI) in (IX) and solve the integral.
Let
Here, the first integral is trivially zero; and the second is
The expectation value of
Substitute
The expectation value
The expectation value
Where,
Write
Use
The values of
Conclusion:
Therefore, the expectation values and
(e)
Whether the uncertainty principle holds or not.
(e)
Answer to Problem 2.42P
The uncertainty principle holds for the given case.
Explanation of Solution
The values of
The product of
Use the expression for
Thus, uncertainty principle holds for this case.
Conclusion:
Therefore, the uncertainty principle holds for the given case.
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Chapter 2 Solutions
Introduction To Quantum Mechanics
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