Which of the following statements is true? I. Every one-particle Hamiltonian operator commutes with L2 and with L₂. II. For a system of n noninteracting particles, each stationary-state wave function has the form V = V1 (91) + V2 (92) +…+¥n (ªn) III. The energy of a system of noninteracting particles is the sum of the energies of the individual particles, where the energy of each particle is found by solving a one- particle Schrödinger equation.

23 An electron in a one-dimensional infinite potential well of length L has ground-state energy E₁. The length is changed to L' so that the new ground-state energy is E₁ = 0.800E₁. What is the ratio L/L?

3. Prove that angular momentum of a particle confined to the xy palne is quantized 4. Neutrons have no electric charge, so they do not interact with the atomic electrons. Hence, they are very penetrating (e.g.,typically 10 cm in lead). Neutron diffraction was proposed in 1934, to exploit de Broglie's hypothesis about the wave nature of matter. Calculate the momentum and kinetic energy of a neutron whose wavelength is comparable to atomic spacing (1.8×1010 m)

1. The angular portions of the wave functions for the px and dxz orbitals of the hydrogen atom are linear combinations of the spherical harmonics and are given by the expressions below. 3 YP. = √(x²+x¹) = -sin coso 4T Wd₂ = 1/2 (Y/² + X/₂ ¹) = eigenfunctions of 15 4π These combinations are chosen to yield real functions. Are these functions a Ĺ sincoscoso -iħ. do ?