Quantum Chemistry Mock Tests

The number of microstates associated with a p² electron configuration is:

The variation method is applied to a trial wavefunction for the helium atom. If the trial wavefunction uses an effective nuclear charge Z_eff as a variational parameter, the optimized Z_eff is found to be approximately 1.69 (instead of Z = 2). This result indicates:

In time-independent perturbation theory, a hydrogen atom is placed in a uniform electric field E along the z-axis (Stark effect). The perturbation Hamiltonian is H' = eEz = eEr cos θ. Which of the following statements about the first-order energy correction is correct for the ground state (1s) of hydrogen? (Recall that the 1s wavefunction is spherically symmetric: ψ₁ₛ = (1/√πa₀³) e^(−r/a₀), where a₀ is the Bohr radius.)

In first-order perturbation theory, if Ĥ = Ĥ⁰ + λĤ' where Ĥ⁰ is the unperturbed Hamiltonian and Ĥ' is the perturbation, the first-order correction to the energy Eₙ⁽¹⁾ is given by:

The particle in a one-dimensional box of length L has energy levels E_n = n²h²/(8mL²). If an electron is confined in a box of length 1.0 nm, the energy difference between the n = 1 and n = 2 states is closest to: (h = 6.626 × 10⁻³⁴ J·s, m_e = 9.109 × 10⁻³¹ kg)

For a particle confined in a one-dimensional box of length L with infinite potential walls, the energy of the n = 3 state relative to the ground state energy E₁ is: (Here, n is the quantum number and E₁ is the ground state energy.)

For the hydrogen atom, the radial wavefunction R_{nl}(r) depends on the principal quantum number n and the azimuthal quantum number l. How many radial nodes does the 3d orbital (n = 3, l = 2) have? (The number of radial nodes is given by the formula: radial nodes = n − l − 1.)

The variation method is applied to a hydrogen-like atom using a trial wavefunction ψ_trial = e^{-αr}, where α is a variational parameter. The optimal value of α that minimizes the energy is: