Concept explainers
(a)
The energy of the state.
(a)
Answer to Problem 47CP
The energy of the state will be
Explanation of Solution
Write the Schrodinger equation for a simple harmonic oscillator.
Here,
Write the given wave function.
Differentiate the above equation with respect to
Differentiate again the above equation.
Conclusion:
Rewrite the Schrodinger equation by substituting the equations for
Compare the coefficients of
Therefore, the energy of the state will be
(b)
The position at which there is no any possibility to find a particle.
(b)
Answer to Problem 47CP
Particle cannot be found at
Explanation of Solution
The possibility of finding a particle depends on the value of wave function. A particle cannot be found at a position where wave function vanishes. The wave function vanishes at
Therefore, the particle cannot be found at
(c)
The most probable position for finding the particle.
(c)
Answer to Problem 47CP
The maximum probability for particles are at
Explanation of Solution
Point of maximum probability is a point of maxima.
Write the condition to be satisfied at the point of maximum probability of finding the particle.
Conclusion:
Substitute
Rewrite the above relation in terms of
Therefore, the maximum probabilities for particles are at
(d)
The value of
(d)
Answer to Problem 47CP
Explanation of Solution
Write the normalization condition.
Conclusion:
Substitute
Rewrite the above relation in terms of
Therefore,
(e)
The classical probability of finding the particle in the width
(e)
Answer to Problem 47CP
Classical probability is zero in the width
Explanation of Solution
Write the equation for potential energy.
Here,
Conclusion:
Substitute
The total energy of oscillator is
Therefore, the classical probability is zero in the width
(f)
The real probability of finding the particle in the width
(f)
Answer to Problem 47CP
The real probability in width
Explanation of Solution
Write the equation to calculate the real probability.
Here,
Conclusion:
Substitute
Rewrite the above relation by substituting
Therefore, the real probability in width
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Chapter 40 Solutions
Physics for Scientists and Engineers with Modern Physics
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