HW7-sol - PHYSICS 4455 QUANTUM MECHANICS Problem Set 7...

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PHYSICS 4455 — QUANTUM MECHANICS Problem Set 7 — Solutions. The finite potential well in one dimension. Consider a structureless, non-relativistic particle of mass m in one dimension, moving in the presence of a finite square well potential: V ( x ) = V 0 | x | < L 0 | x | ≥ L . Just like the semi-infinite square well, there are bound and scattering states associated with this one, naturally for E < 0and E > 0, respectively. We assume that V 0 > 0. 1. Let’s find the energies and wavefunctions for all the bound states. Let’s divide the real axis into three regions: Region (1) with x ≤− L , region (2) with L < x < L , and region (3) with x L . In regions (1) and (3) the Schrödinger equation takes the form: 2 2 m ϕ  ( x )= E ϕ( x ) (1) and (3) andinregion(2)wehave 2 2 m d 2 dx 2 V 0 ϕ( x )= E ϕ( x ) (2) It is natural to assume that E cannot be smaller than V 0 . Looking for bound states, we will restrict ourselves to the energy range V 0 < E < 0. We also introduce λ 2 =− 2 m 2 E > 0and k 2 = 2 m 2 ( V 0 + E ) > 0 both of which are confined to the range 0 2 , k 2 < 2 m 2 V o For future reference, let us relate λ and k : λ= 2 m 2 V o k 2 So, rewriting the Schrödinger equation, we have 0 = d 2 dx 2 −λ 2 ϕ( x )= 0 (1) and (3) which has solutions e ±λ x and 0 = d 2 dx 2 + k 2 ϕ( x )= 0( 2 ) which has solutions e ± ikx (note that we need only the positive roots, λ=+ − E ,and k =+ V 0 + E ). For bound states, we demand that the wave function vanishes at x →±∞
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HW7-sol - PHYSICS 4455 QUANTUM MECHANICS Problem Set 7...

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