Question

2. At time t=0 the wave function for hydrogen atom is ψ(r,t = 0) = 1/√10(2ψ₁₀₀ + ψ₂₁₀ + √2ψ₂₁₁ + √3ψ₂₁₋₁)

where the subscripts are values of the quantum numbers n,t ,m. Ignore spin and radiative transitions. (a) What is the expectation value for the energy of this system ? (10%) (b) What is the probability of finding the system with t=1, m=+1 as a function of time ? (10%) (C) What is the probability of finding the electron within 10-10 cm of the proton at time t=0 ? (You can use mathematical tool/program to help you.) (15%) (d) The wave function evolves in time t ? (10%) (e) Suppose a measurement is made which shows L2 = 2h2 and Lz=ħ. What is the wave function immediately after such a measurement ? (10%)

Answer 1

The question pertains to the properties of a hydrogen atom's wave function and includes finding expectation values for energy, time-dependent probabilities, and post-measurement wave function characteristics.

Step-by-step explanation:

The question asks about the expectation value for the energy, the probability of finding a specific quantum state over time, the probability of finding an electron close to the proton at t=0, how the wave function evolves over time, and the wave function after a particular measurement of angular momentum for a hydrogen atom.

(a) The expectation value for energy is the sum of the probabilities of each state multiplied by their respective energies, given by E = Σ |c_i|^2 E_i, where c_i are the coefficients and E_i the energy levels corresponding to the quantum states.

(b) To find the probability of the electron being in the state with l=1 and m=+1, we consider the coefficient of ψ_211 in the initial wave function, which will give the probability as a constant since the potential is time-independent.

(c) The probability of finding the electron within a certain region is found by integrating the probability density |psi|^2 over that region, which will require computational tools or physics software.

(d) The wave function evolves over time according to the time-dependent Schrödinger equation, but for a time-independent potential, this evolution is simpler, with each state acquiring a phase factor e^(-iE_nt/h).

(e) When a measurement confirms that L^2 = 2h^2 and Lz= ħ, the wave function collapses to the state that corresponds to those values of angular momentum, which is ψ_21+1 in this case.

Answer 2

The energy expectation is computed by summing probabilities of states multiplied by their respective energies, while calculating the probability at t=1, m=+1 entails using the time evolution operator.

Step-by-step explanation:

The expectation value of energy involves computing the integral of the Hamiltonian operator with respect to the given wave function at t=0. Utilizing the given wave function, each term represents a different state of the hydrogen atom.

The probability amplitude associated with each state is squared and multiplied by the energy of the corresponding state to compute the expectation value. This is done by applying the integral formula for expectation values in quantum mechanics. Each term's contribution to the overall expectation value is calculated by considering the probability amplitude and energy of the respective state.