University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

341

Pages:

7
Author:

Jabin F.

Unsymmetric poth well V(x) = 0 if x < 0 V₀ otherwise Ψₙ(x) = { Asin(kx) if x > 0 Be^(κx) if x < 0 } Eₙ = (ħ²k²)/(2m) = - (ħ²κ²)/(2m) <Ψₙ|Ψₘ> = ∫₀∞ Ψₙ*(x)Ψₘ(x) dx = δₙₘ Ψ(x) = { Asin(kx) if 0 < x < L 0 otherwise } 1 = ∫|Ψ(x)|² dx = A² ∫₀ˡ sin²(kx) dx = A² L/2 E = <Ψ|Ĥ|Ψ>/<Ψ|Ψ> = (ħ²/2m) ∫₀ˡ Ψ*(x) d²Ψ/dx² dx = (ħ²k²)/(2m) Ψ(0) = Ψ(L) = 0 Eₙ = (n²ħ²π²)/(2mL²) For nth state n²/2 • Probability P = ∫|Ψₙ(x)|² dx = 1 0 p = ∫|Ψₙ(x)|² dx = ½ a/2 Problem 7th same as previous probability \(p=\frac{1}{4}-\frac{1}{2\pi}\) \(Ot\circ\frac{L}{4}\) 80146 For first excited state, \(n=\alpha^{\prime}\) 42(1) = ↑ Sin 24 L otherwise 0, 4 most probable value \[\frac{d}{dm}|\psi_{2}(on)|^{2}=0\] \frac{d}{dx}|\psi_{2}(x)|^{2}=0 sin(\frac{2\pi x}{L})cos(\frac{2\pi x}{L})=0 sin(4\pi x/L)=0=sin(m\pi) L for value of m→ π=0,\frac{L}{4},\frac{L}{2},\frac{3L}{4},L \frac{4\pi x}{L}=m\pi x=\frac{mL}{4} \frac{d^{2}}{dx^{2}}|\psi_{2}(x)|^{2}>0 for x=\frac{L}{2} \frac{d^{2}}{dx^{2}}|\psi_{2}(x)|^{2}<0 for x=\frac{L}{4},\frac{3L}{4} The equations cover the wavefunction, normalization, energy eigenvalue, boundary conditions, quantization, normalized wavefunction, and energy levels. The text is neatly written in black ink on a white background. The equations are aligned vertically and horizontally, enhancing readability. The equations involve variables A, B, k, and n, along with trigonometric functions sine and cosine. There are also numerical values and symbols such as parentheses and equal signs. The equations are arranged vertically, with some horizontal alignment. The handwriting is clear and legible, although some symbols might be ambiguous due to the handwriting style. B≠0, A=0 cos(ka/2) = 0 ka/2 = (2n+1)π/2 k = (2n+1)π/a Ψ(x) = Bcos((2n+1)πx/2a) = 1,3,5-009 Blos (mr), m=1,3,5- a A Sin(n), n= 2,4,6.--- even otherwise 0 \[\frac{-a}{2}\le x\le\frac{+a}{2}\] Ψη(m) = for particular value of n \[|B|^{2}=|A|^{2}=\frac{2}{a}\] \[\int_{-\infty}^{\infty}|\psi_{n}(x)|^{2}dx=1\] \[\psi_{n}(\eta)=&(\begin{matrix}\int\frac{2}{a}css(\frac{n\pi\pi}{a}),&n=&i,s_{i}-\cdot\cdot\\ &i\cdot\cdot&-\frac{a}{2}\le\eta\le\frac{+d}{2}\end{matrix}\]