
Concept explainers
(a)
To show that the first term in the Schrodinger equation reduces to the kinetic energy of the quantum particle multiplies by the wavefunction for a freely moving particle with the wave function
(a)

Answer to Problem 54P
It is showed that the first term in the Schrodinger equation reduces to the kinetic energy of the quantum particle multiplies by the wavefunction for a freely moving particle with the wave function
Explanation of Solution
Write the Schrodinger’s equation.
Here,
Write the statement to be proved.
Here,
Write the expression of the given wavefunction.
Here,
Put equation (III) in equation (II).
Take the derivative equation (III) with respect to
Take the derivative of the above equation with respect to
Put equations (V) in the left-hand side of equation (II) and rearrange it.
Write the equation for the reduced Planck’s constant.
Here,
Write the equation for the wave vector.
Here,
Put equation (VII) and (VIII) in (VI).
Write the equation for the de Broglie wavelength.
Here,
Rewrite the above equation for
Put the above equation in equation (IX).
Write the equation for kinetic energy.
Put the above equation in equation (XI).
Conclusion:
Equation (XIII) is exactly the same as equation (IV) which has to be proved.
Thus, it is showed that the first term in the Schrodinger equation reduces to the kinetic energy of the quantum particle multiplies by the wavefunction for a freely moving particle with the wave function
(b)
To show that the first term in the Schrodinger equation reduces to the kinetic energy of the quantum particle multiplies by the wavefunction for a particle in a box with the wave function
(b)

Answer to Problem 54P
It is showed that the first term in the Schrodinger equation reduces to the kinetic energy of the quantum particle multiplies by the wavefunction for a particle in a box with the wave function
Explanation of Solution
Write the expression of the given wavefunction.
Put equation (XIV) in equation (II).
Take the derivative equation (XIV) with respect to
Take the derivative of the above equation with respect to
Put the above equation in the left-hand side of equation (XV) and rearrange it.
Put equation (VII) and (VIII) in the above equation.
Put equation (X) in the above equation.
Put equation (XII) in the above equation.
Conclusion:
Equation (XVI) is exactly the same as equation (XV) which has to be proved.
Thus, it is showed that the first term in the Schrodinger equation reduces to the kinetic energy of the quantum particle multiplies by the wavefunction for a particle in a box with the wave function
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Chapter 28 Solutions
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