
(a)
To Write: The formula for the total zero-point energy of all the modes of the field inside the box in terms of a triple integral over the mode numbers in the x, y, and z direction.
(a)

Answer to Problem 24P
Explanation of Solution
Formula used:
Total energy within the box can be written as:
Where,
Factor of
Calculation:
Wavelength can be given as:
So, speed of light:
So,
From equation (1)
Conclusion:
The formula for the total zero-point energy of all the modes of the field inside the box is
(b)
To Show: The Planck length indeed has units of length.
To Calculate:Numerical value of Planck length.
(b)

Answer to Problem 24P
Explanation of Solution
Given:
Planck Length =
Formula used:
Dimension formula of
Calculation:
Dimension formula for given expression can be calculated as:
Planck length can be calculated as:
Conclusion:
It is proved that the Planck length indeed has units of length and its magnitude is
(c)
To Estimate: The energy per unit volume in empty state and equivalent mass density.
(c)

Answer to Problem 24P
Explanation of Solution
Formula used:
Where,
Calculation:
But,
Therefore,
Now, mass density can be calculated as
It is much greater than the average mass density of ordinary matter in the universe.
Conclusion:
Average mass density is
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Chapter A Solutions
An Introduction to Thermal Physics
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