Consider the wave function V222(x, ỹ, z) for a particle in a cubic box. Figure 4.45a shows a contour plot in a cut plane at ž = 0.75. (a) Convince yourself that the contour plot in a cut at ž = 0.25 would have the same pattern, but each positive peak would become negative, and vice versa. (b) Describe the shape of this wave function in a plane cut at ỹ = 0.5.

What is the value of the commutator [P , â4 ].

1:51 PM Sun Oct 9 -1/2 5) For a particle on a ring with the wavefunction: () = π kinetic energy of the particle using the operator K = -1/2 Y(0) = IT 21T K=-ħ²d² 21d² 2 cos ($) = kdo T (17¹/2)² 211 - f = "Cos (0) . # de TỈ Cos (O). t d. TỈ Cosco) do = 21 0 21 cos() calculate the average cos(0). -ħ² d² do 21 do² 10:52 @ 79% + : 5 9 +

Consider a particle trapped in a iD box with zero potential energy with walls at x = 0 and x = L. The general wavefunction solutions for this problem with quantum number, n, are: yn(x) = The corresponding energy (level) for each wavefunction solution is: n?h? En 8ml? a) Is the particle always uniformly distributed throughout the box? Explain your answer. b) Consider the molecules: CH2=CH-CH=CH-CH=CH-CH=CH-CH=CH2. Follow from Ex 4.7 in the textbook, let's assume that the 1o electrons that make up the double bonds can exist everywhere along the carbon chains. The electrons can then be considered as particles in a box; the ends of the molecule correspond to the boundaries of the box with a finite or zero potential energy inside. In this "molecular box", 2 electrons can occupy an energy level. What are quantum states that the electrons from this molecule can occupy in the ground state? What's the smallest frequency of light that can excite the electron? Briefly explain why. Note that the…