Review schrodinger equations not dependent on 3D time in ball coordinates v² + V(F) )p(F) = E Þ(F) 2m With = (r, 0,9) and E is the energy system. Assume the potential is only radial function V(*) = V(r) To solve the Schrodinger equation above apply the method p(r, 0, q) = R(r)P(0)Q(9) variable separation problem : By defining u(r) = rR(r) where u(r) qualifies the limit of u(0) = 0 and u(--) = Show that fulfilling the radial equation u(r) h? d?u h2 l(1 + 1) +V(r) + 2m u = E u 2m dr2 r2

Show that ? (x,t) = A cos (kx - ?t) is not a solution to the time-dependent Schroedinger equation for a free particle [U(x) = 0].

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

consider an infinite square well with sides at x= -L/2 and x = L/2 (centered at the origin). Then the potential energy is 0 for [x] L/2 Let E be the total energy of the particle. =0 (a) Solve the one-dimensional time-independent Schrodinger equation to find y(x) in each region. (b) Apply the boundary condition that must be continuous. (c) Apply the normalization condition. (d) Find the allowed values of E. (e) Sketch w(x) for the three lowest energy states. (f) Compare your results for (d) and (e) to the infinite square well (with sides at x=0 and x=L)