QM in 1D position-space

QM of particle in a 1D finite square well

A particle in the finite square well potential

where

has odd bound states

where

and even bound states

where

Proof

Bound states correspond to . llows that . For the Schrödinger equation becomes

where . Thus for , and applying we conclude . For the Schrödinger equation is

where . Thus for . For odd solutions, , hence

thus, by continuity we have and by smoothness we have . Thus Let and . Since , it follows , hence

which may be solved numerically. A similar treatment for the even case¹ gives the result stated above.

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Footnotes

1. 2018. Introduction to quantum mechanics, §2.6, p. 72 ↩