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
^{1}H^{127}I. 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
^{1}H^{35}Cl (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 ^{16}O-^{16}O bond is 1142 N.m^{-1}, estimate the zero-point energy in the O_{2} bond. Explain your assumptions. What fraction of the 5.15 eV bond dissociation energy is the zero-point energy for O_{2}?
Ans1 is given below:C) Ground state function and first excited state function are given as below,
It is the condition of orthogonality of two wave functions.
Q 1: For particle in a box problem, answer the following questions, a) Why n=0 is...
this is statistical mechanics 4. Calculate the probability that at 300 K a given particle in a 2D square box with a length of 1 cm is found in a) the quantum state (nx, ny) (3,1). b) the energy level e. 4. Calculate the probability that at 300 K a given particle in a 2D square box with a length of 1 cm is found in a) the quantum state (nx, ny) (3,1). b) the energy level e.
help Part B. Open questions. 1. (30 points) For the one-dimensional particle in a box of length L. a. Write the wavefunction for the fifth excited state b. Calculate the energy for the fifth excited state when L = 18 and m = Ing. c. Write an integral expression for the probability of finding the particle between L/4 and L/2, for the second excited state. d. Calculate the numerical probability of finding the particle between 0 and L15, for the...
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harmonic oscillator potential with natural frequency W.Att-0 an external magnetic field is turned on which is uniform in space but oscillates with temporal frequency W as follows. E(t)-Bo sin(at) At time t>0, the perturbation is turned off. Assuming that the system starts off at t-0 in the ground state, apply time-dependent perturbation theory to estimate the probability that the system ends up in an...
Item 5 An particle in a 1-D box of length L=2A∘ has allowed energy levels that include 84.82 eV and 150.79 eV; however, the quantum numbers for these two states are unknown. Part A - What is the ground-state energy of the system? E1=9.42eV E1=37.7eV E1=5.30 eV E1=16.75 eV E1=28.3eV E1=21.2 eV Part B - What is the de Broglie wavelength for the n=2 state? 2.5A∘ 0.5A∘ 1.5A∘ 3.0 A∘ 1.0 A∘ 2.0 A∘
Quantum Mechanics Problem 1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
Griffiths Introductory to Quantum Mechanics (3rd Edition): Problem 7.9 Problem 7.9 Consider a particle of mass m that is free to move in a one- dimensional region of length L that closes on itself (for instance, a bea that slides frictionlessly on a circular wire of circumference L, as Problem 2.46) (a) Show that the stationary states can be written in the form 2π inx/L where n 0, t l, 2, , and the allowed energies are In Notice that-with...
Answer all questions please 5. Consider a particle in the first excited state ofa rigid box of length a. (a) Find the probability density (b) where is the particle most likely to be found? 6. Determine the wavelength of the photon emitted when an electron in a hydrogen atom makes transition from the 5 excited state to the following states (a) ground state (b) 1 excited states (c) 2 excited state Determine whether the emission is visible, uv or infrared...
11. Use a one dimensional particle in a box model for the nucleus to answer the following questions. a. Explain why a nucleus with two neutrons and two protons has less kinetic energy than a nucleus with 3 neutrons and one proton. b. Why do nuclei with large numbers of nucleons have an excess of neutrons over protons? 11. Use a one dimensional particle in a box model for the nucleus to answer the following questions. a. Explain why a...
Quantum, 1D harmonic oscillator. Please answer in full. Thanks. Q3. The energy levels of the 1D harmonic oscillator are given by: En = (n +2)ha, n=0. 1, 2, 3, The CO molecule has a (reduced) mass of mco = 1.139 × 10-26 kg. Assuming a force constant of kco 1860 N/m, what is: a) The angular frequency, w, of the ground state CO bond vibration? b) The energy separation between the ground and first excited vibrational states? 7 marks] The...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...