Tutors: Kurur/Pandey

1. (a) Calculate the energy levels for n = 1, 2, and 3 for an electron in an infinite potential well of width 0.25 nm. (b) If an electron makes a transition from n = 2 to n = 1 what will be the wavelength of the emitted radiation?
2. (a) Evaluate the probability of locating a particle in the middle third of 1-D box. (b) Find the probability that a particle in a box L wide can be found between x = 0 and x = L/n when it is in the nth state.
3. Consider two wave functions which describe any two different states of a particle in a box. Show that these are orthonormal.
4. Verify the uncertainty principle for the particle in a box.
5. Discuss the source and nature of degeneracy for a particle in a 2−D and 3−D box.
6. A particle is confined to a two dimensional box of length L and 2L. What are the allowed energy levels?
7. Many proteins contain metal porphyrin molecules. These molecules are planar and contain 26 π electrons. If the length of the molecule is ∼ 1000 pm, then what is the predicted lowest energy absorption of the porphyrin molecule?
8. Consider a particle confined to move in the region −a/2 ≤ x ≤ a/2 and whose wavefunction is Ψ(x, t) = cos(π x/a) exp(−iω t). [8 × 5]
   1. Find the potential V(x) and hence write the Hamiltonian.
   2. What is the probability of finding the particle in the region −a/6 ≤ x ≤ a/6?
   3. What is the expectation for the energy of the particle in this state?
   4. What is the uncertainty in the energy measurement?
   5. An operator P is defined as P ψ(x) = ψ(−x). Show that P is an acceptable quantum mechanical operator.
   6. Is Ψ(x,t) an eigenfunction of P? If so, what is its eigenvalue?
   7. Decide whether P commutes with the Hamiltonian.
   8. If another wavefunction of the particle is Ψ′(x, t) = sin(2 π x / a) exp(−i ω′ t), determine whether Ψ(x, t) and Ψ′(x, t) are orthogonal.

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