ETS2017 Problem 13

[GeorgesOatesLarsen]

A particle of mass m is confined inside a one-dimensional box (infinite square well) of length a. The particles ground state energy is which of the following?

(A) h_bar/8ma (B) h_bar^2/8ma^2 (C) h_bar^2/8ma^2 (D) h_bar^2pi^2/8ma^2 (E) h_bar^2a^2/8mc^2

[GeorgesOatesLarsen]

This seems like a good place to start developing methods for creating generalized relational problems -- IE, starting from EITHER a trivia equational OR computational problem, and generating variants of both types.

In particular, it seems worth it to apply the following generalizations:

• Trivia.Equational/Ratio/Value
• Variants for first three energy states/wavenumbers.
• Variants for unfixed states/wavennumbers.

This will result in around six major variants, and we'll basically have infinite square well completely covered for the purposes of practice.

Might be worth it to see if this generalizes also to finite square well, although those tend to have take on a pretty distinct form due to their nature and the added edge cases. Not to mention, ETS2017 has other problems covering this, so we can just re-do the process based on those, so maybe not worthwhile here.

[GeorgesOatesLarsen]

Generalized expression, here, is E_n = h_bar * omega_n

The wavefunction is in the form of a free particle inside the well, due to the zero potential therewithin.

The angular frequency and spatial frequency of free-particle waves are inextricably linked by Schrodinger's equation. Substituting the free-particle wave into schroedinger's time dependent equation will yield the relation:

omega = h_bar * k^2/2m

Therefore:

E = h_bar^2 * k^2/2m

where k is the angular spatial frequency.

The spatial half-wavelength of the particle must divide the length, a, of the box.

Essentially this yields the constraint: pi * n/L = k

So, we arrive at:

E = h_bar^2 pi^2 n^2/(2 L^2 m)

[GeorgesOatesLarsen]

Completed. Will close with next push. Need to finish ETS17P6 for stability