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How do you use the Schrodinger equation to determine a particle’s position in free space?

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2. A wave function is a linear combination of 1s, 2s, and 3s orbitals: y(r) = N(0.25 y,s + 0.50W2, +0.30W). Find the normalization constant N, knowing that 1s, 2s, and 3s orbitals are normalized.
Review schrodinger equations not dependent on 3D time in ball coordinates -V² + V(f) ) µ(F) = E Þ(*) 2m F = (r,0,9) and E is the energy system. Assume the potential is only radial function V (f) = V(r). With %3D 'To solve the Schrodinger equation above apply the method ψη, θ φ) -R(r)P (θ) Q (φ) variable separation problem : By defining the separation constant in the angular function 0 as (1 + 1) Also show that P(0) = P"(cos 0) angular function solution P(0) can be written as Polynomial Associated Legendre The 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
18. In differential form, what is the kinetic energy operator equal to? Does this depend on the problem that you are solving in quantum mechanics? What is the eigenvalue spectrum for this operator?
Consider a particle in a 2-D box having Lx = 10 nm and Ly = 10 nm. a) Make a surface plot of all the wave functions for the first and second energy levels. b) What is the degeneracy of the second energy level? Compare and contrast the wave functions of the second energy level. c) How does the number of nodes in the x-coordinate change as n increases? How does the number of nodes in the y-coordinate change as n, increases? d) Explain whether or not those same states would be degenerate if Lx = 10 nm and Ly = 15 nm.
3. Plane waves and wave packets. In class, we solved the Schrodinger equation for a "free particle" (e.g. when U(x,t) = 0). The correct[solution is (x, t) = Ae(px-Et)/ħ This represents a "plane wave" that exists for all x. However, there is a strange problem with this: if you try to normalize the wave function (determine A by integrating * for all x), you will find an inconsistency (A has to be set equal to 0?). This is because the plane wave stretches to infinity. In order to actually represent a free particle, this solution needs to be handled carefully. Explain in words (and/or diagrams) how we can construct a "wave packet" from the plane wave solution. (Hint 1: consider a superposition of plane waves for a limited range of momentum/energy. Hint 2: have a look at the brief discussion in the middle of pg. 278 and especially pg. 308-309 of the text.)
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
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