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What can you say about the relationship between the density at the origin and the maximum probability mass in a thin shell, as a function of the dimension \(d\)?

Answer :

In higher dimensions, the relationship between the density at the origin and the maximum probability mass in a thin shell becomes more pronounced.

The probability density function of a particle in d dimensions can be represented as a radial function multiplied by a spherical harmonics function.

As the dimension d increases, the volume of the unit sphere (the thin shell) becomes more concentrated near the equator and the maximum probability mass becomes more pronounced.

This means that in higher dimensions, the probability density function is more likely to have a sharp maximum at a point outside the origin, and a lower density at the origin. This phenomenon can be seen in various physical systems, such as the wave functions of electrons in quantum mechanics.

Therefore, the relationship between the density at the origin and the maximum probability mass in a thin shell is dependent on the dimensionality of the system.

Learn more about dimensions here.

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