QM in 1D position-space

QM of particle in a 1D finite square well

A particle in the finite square well potential

𝑉(𝑥)={−𝑉0𝑥∈[−𝑎,𝑎]0elsewhere

where

𝜅=√−2𝑚𝐸ℏ=√𝑧20+𝑧2𝑎𝑙=√−2𝑚(𝐸+𝑉0)ℏ=𝑧𝑎

has odd bound states

𝜓(𝑥)=⎧{ {⎨{ {⎩𝐴𝑒𝜅𝑥𝑥<−𝑎𝐹sin⁡𝑙𝑥𝑥∈[−𝑎,𝑎]−𝐴𝑒−𝜅𝑥𝑧=𝑥>𝑎

where

𝐴=−𝐹𝑒𝜅𝑎sin⁡𝑙𝑎cot⁡𝑧=√𝑧20𝑧2−1

and even bound states

𝜓(𝑥)=⎧{ {⎨{ {⎩𝐴𝑒𝜅𝑥𝑥<−𝑎𝐹cos⁡𝑙𝑥𝑥∈[−𝑎,𝑎]𝐴𝑒−𝜅𝑥𝑧=𝑥>𝑎

where

𝐴=𝐹𝑒𝜅𝑎cos⁡𝑙𝑎tan⁡𝑧=√𝑧20𝑧2−1

Proof

Bound states correspond to −𝑉0 ≤𝐸 <0. llows that 𝜓( −𝑥) = ±𝜓(𝑥). For 𝑥 ∈( −∞,𝑎) the Schrödinger equation becomes

𝐸𝜓(𝑥)=−ℏ22𝑚𝑑2𝑑𝑥2𝜓(𝑥)𝑑2𝑑𝑥2𝜓(𝑥)=−2𝑚𝐸ℏ2𝜓(𝑥)=𝜅2𝜓(𝑥)

where 𝜅 =√−2𝑚𝐸/ℏ. Thus 𝜓(𝑥) =𝐴𝑒−𝜅𝑥 +𝐵𝑒𝜅𝑥 for 𝑥 ∈( −∞,𝑎), and applying lim𝑥→−∞𝜓(𝑥) =0 we conclude 𝐴 =0. For 𝑥 ∈[ −𝑎,𝑎] the Schrödinger equation is

𝐸𝜓(𝑥)=−ℏ22𝑚𝑑2𝑑𝑥2𝜓(𝑥)−𝑉0𝜓(𝑥)𝑑2𝑑𝑥2𝜓(𝑥)=−2𝑚(𝐸+𝑉0)ℏ2𝜓(𝑥)=−𝑙2𝜓(𝑥)

where 𝑙 =√−2𝑚(𝐸+𝑉0)ℏ. Thus 𝜓(𝑥) =𝐹sin⁡𝑙𝑥 +𝐺cos⁡𝑙𝑥 for 𝑥 ∈[ −𝑎,𝑎]. For odd solutions, 𝜓( −𝑥) = −𝜓(𝑥), hence

𝜓(𝑥)=⎧{ {⎨{ {⎩𝐴𝑒𝜅𝑥𝑥<−𝑎𝐹sin⁡𝑙𝑥𝑥∈[−𝑎,𝑎]−𝐴𝑒−𝜅𝑥𝑧=𝑥>𝑎𝜓(−𝑥)=⎧{ {⎨{ {⎩𝐴𝜅𝑒𝜅𝑥𝑥<−𝑎𝐹𝑙cos⁡𝑙𝑥𝑥∈[−𝑎,𝑎]𝐴𝜅𝑒𝜅𝑥𝑥>𝑎

thus, by continuity we have 𝐴𝑒−𝜅𝑎 = −𝐹sin⁡𝑙𝑎 and by smoothness we have 𝐴𝜅𝑒−𝜅𝑎 =𝐹𝑙cos⁡𝑙𝑎. Thus 𝜅 = −𝑙cot⁡𝑙𝑎 Let 𝑧 =𝑙𝑎 and 𝑧0 =𝑎ℏ√2𝑚𝑉0. Since 𝜅2 +𝑙2 =2𝑚𝑉0ℏ2, it follows 𝜅𝑎 =√𝑧20−𝑧2, hence

cot⁡𝑧=−𝜅𝑎𝑙𝑎=√𝑧20−𝑧2𝑧=√𝑧20𝑧2−1

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

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tidy | en | SemBr

Footnotes

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