Question

Q 1: For particle in a box problem, answer the following questions,
a) Why n=0 is not an allowed quantum number?
b) En = 0 is not allowed for particle in a box, why?
c) Ground state wavefunction is orthogonal to the first excited state wavefunction, what
does it mean?

Q 2: An electronic system that is treated as particle in 3-D box with dimensions of 3Å x 3Å x 4Å. Calculate the wavelength corresponding to the lowest energy transition if the system has 18 electrons.

Q 3: Use the ground state wavefunction of particle-in-a-box and prove that the momentum operator is Hermitian.

Q 4: What are the differences between the conclusions for a harmonic oscillator drawn by classical mechanics and quantum mechanics?

Q 5: The fundamental frequency is 2230 cm^-1 for ¹H¹²⁷I. If the H atom is replaced with D (isotope of H atom), calculate the fundamental frequency assuming that the force constant stays the same. Repeat the calculation if the diatomic molecule under consideration of ¹H³⁵Cl (fundamental frequency = 2886 cm^-1). Explain your observations. [Do not use any other information other than what has been given in the question.]

Q 6: Given the force constant of ¹⁶O-¹⁶O bond is 1142 N.m^-1, estimate the zero-point energy in the O₂ bond. Explain your assumptions. What fraction of the 5.15 eV bond dissociation energy is the zero-point energy for O₂?

Answer 1

Ans1 is given below:
(as The wave function for particle in a boxu given as, - 4 - 2 sin nare where 4 = wave function l = length of the box N = quantum no. Uz position of e in a coordinate If n=D, then sin (102) will become zero, * sin (max) = &in (0294) = 0 [11n6-o] Due to which, 4 =0 Hence, the wave function will become zero, or the probability of finding e in a box becomes zero which is not possible.

For particle in a box, energy is given as below, En = n²h 8me² where, En = energy of electron in level na quantum state of é h = Planck's constant e = length of the box m = mass of é If n o , then En = 0 It shows that energy of ē is zero. As well as the momentum of é is zero. (n = 0, E=0) contradict the existence of electron
C) Ground state function and first excited state function are given as below,

It is the condition of orthogonality of two wave functions.

Ground state function, (n =) Eisst excited state function (n=2) tot e infor) 2 sin ( 22 ) da 2 / (sin x) (sin 28x) dx On 4 y solving, = 0