Review schrodinger equations not dependent on 3D time in ball coordinates-V² + V(f) ) µ(F) = E Þ(*)2mF = (r,0,9) and E is the energy system. Assume the potential is only radial functionV (f) = V(r).With%3D'To solve the Schrodinger equation above apply the methodψη, θ φ) -R(r)P (θ) Q (φ)variable separationproblem : By defining the separation constant in the angular function 0 as (1 + 1) Also show thatP(0) = P"(cos 0)angular function solution P(0) can be written as Polynomial Associated LegendreThe combined angular functions of the sphere are known as the "harmonic function of the sphere"(spherical harmonics)Y(0, p) x Pi" (cos 0)e±imp

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Consider the 1D time-independent Schrodinger equation ħ² ď² 2m dr² with the potential where to is a parameter. (a) Show that V(x) = +V(x)] = Ev v is a solution of the Schrodinger equation. ħ² mx² sech² 1 = A sech x xo (₁)
A particle inside an infinite square well ( a = 1 ) start at the initial state Y(x, 0) = v3(1 – x)0 < x < 1 a) calculate the coefficient C1 b) what does |cq]² mean physically ?
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Complete the derivation of E = Taking the derivatives we find (Use the following as necessary: k₁, K₂ K3, and 4.) +- ( ²) (²) v² = SO - #2² - = 2m so the Schrödinger equation becomes (Use the following as necessary: K₁, K₂, K3, ħ, m and p.) 亢 2mm(K² +K ² + K² v k₁ = E = = EU The quantum numbers n, are related to k, by (Use the following as necessary: n, and L₁.) лħ n₂ π²h² 2m √2m h²²/0₁ 2m X + + by substituting the wave function (x, y, z) = A sin(kx) sin(k₂y) sin(kz) into - 13³3). X What is the origin of the three quantum numbers? O the Schrödinger equation O the Pauli exclusion principle O the uncertainty principle Ⓒthe three boundary conditions 2² 7²4 = E4. 2m
Review schrodinger equations not dependent on 3D time in ball coordinates -v² + V(f) ) µ(F) = E Þ(*) 2m i = (r, 0, 4) V (f) = V(r). and E is the energy system. Assume the potential is only radial function To solve the Schrodinger equation above apply the method With y(r, 0, q) = R(r)P(0)Q(9) variable separation problem : Specify a common solution (r, 0,0) for l = 0 and V(r) = 0
(x, t) = Ae-iwt e-(mw/ħ).x² which is a solution to Schrödinger's equation. Determine the potential V(x) that is consistent with this wave function, Note: You do not have to normalize V since Schrödinger's equation is linear.