Concept explainers
(a)
The sketches of the wave function of a particle confined in a finite box.
(a)
Answer to Problem 71P
The wave function is given below.
Explanation of Solution
For a particle in a box, the wave function in the
For
Since the wave function is a sinusoidal curve decaying outside the box, the box contains less than half the wavelength. Thus,
i.e.
Here,
For
Since the wave function is a sinusoidal curve decaying outside the box, the box contains less than one wavelength and more than half the wavelength.
i.e.
For
Since the wave function is a sinusoidal curve decaying outside the box, the box contains less than
i.e.
From all the above given wave functions, it can be generalised that
Here,
(b)
Check whether the condition
(b)
Answer to Problem 71P
Yes. The given condition is valid for a particle confined in a finite box.
Explanation of Solution
The energy of
Write the expression for energy of a particle in a infinite box in terms of wavelength
Here,
Rearranging (II) for
Write the expression for ground state energy
Write expression (I)
Taking reciprocal and squaring (I)
While taking reciprocal, the direction of inequality changes.
Multiplying (III) by
Substituting (III) in (VI)
Substituting for
Cancelling out the common terms in numerator and denominator
Thus, the given condition is valid.
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Chapter 28 Solutions
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