The complete step wise calculation of the average potential energy is given as below:-
Consider the quantum mechanical vibration of H2 in the n = 1 state. Calculate the expectation...
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic oscillator is p1(x)- Axe-bx2 where mo 2h and ?1/4 (Note: In last week's homework there was an "h" where there should have been ?. This has been corrected in this week's assignment.) (a) By applying the lowering operator to ),obtain an explicit form for o(x) (i.e. the n-0 wavefunction) (b) By applying the raising operator to x), obtain an explicit form for p2(x) (i.e....
1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement on a particle prepared in the ground state of harmonic oscillator. The formula will depend on h, m andw (b) Estimate order of magnitude of the standard deviation in (a) for the LIGO mirror of mass 10 kg and w 1 Hz. (c) A coherent state lo) is defined to be the eigenstate of the lowering operator with eigenvalue a, i.e. à lo)a) Write la) as where...
= μ = 0.5 This problem deals with the vibrational motion of the H2 molecule (reduced mass- amu). The Hamiltonian for this system is: h2 d 1, e2ndxī + 2kx2. 5 pts] By direct substitution of the wavefunction labelled by the quantum number v, Where k is a constant related to the bond strength. V.(x), in the Schrödinger Equation, show that the wavefunction Ψ(x) = Noe- )' where α = ( corresponds to the ground vibrational state of H2 having...
4. What is the extent of the ground state wavefunction of the vibration (approxi- mated as a harmonic oscillator)for NO. Note that vibrational transitions in NO are observed at 5.6 × 1013 Hz and its overtones. Hint: be careful with what (1 point) effective "mass" vou use for the vibration. 4. What is the extent of the ground state wavefunction of the vibration (approxi- mated as a harmonic oscillator)for NO. Note that vibrational transitions in NO are observed at 5.6...
Example Question Suppose a molecule exists as a one dimensional harmonic oscillator in a superposition state that is given by the following wavefunction: 1 15 Y = -4 +291 Where Y. and Y, are the ground state and the first excited state wavefunctions of the harmonic oscillator. Evaluate the expectation value of the vibrational energy this molecule in such a superposition state (in cm ) given that the vibration constant for the molecule is about 1800 cm
1000 diatomic molecules with vibrational state described by N molecules in the ground vibrational state O molecules in the lowest potential Total Energy Potential Energy state M molecules in an excited vibrational states P molecules in an excited potential energy states Schematic of Energy Eigenfunctions and the 1. Consider a sample of 1000 identically prepared diatomic molecules, each of which can be Potential Energy function of the Harmonic Oscillator described by the ground state of the Harmonic oscillator: Ψ-ψ。. If...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
Solve the following problem: 6. a) Find an expectation value<p> in the n" state of the harmonic oscillator. Hint: it may be useful to recall the following equations: a.y-n+1 1 (Fip+max) /2 hma n+1 and а. b) Compute an expectation value of the kinetic energy in the n" state of the harmonic oscillator. Show that it is exactly half of the total energy in that state
4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The system is in thermal equilibrium at temperature T. The oscillator is described by the following density matrix: A exp kaT where H is the usual harmonic oscillator Hamiltonian and kB is Boltzmann's constant. Working in the Fock (photon number) basis: a. Find the diagonal elements of ρ b. Determine the normalization constant A. c. Calculate the expectation value of energy (E 4. (20 points). Consider a quantum...
In the lecture notes, we only solved the TISE for the quantum harmonic oscillator 1 Now, write down the actual solution of the wavefunction of the quantum harmonic oscillator, i.e. the solution that solves TDSE not TISE. 2. We consider the Quantum Harmonic Oscillator In Heisenberg Picture: (a) Hamiltonian to use is the quantum harmonic oscillator Hamiltonian Solve the Heisenberg equations of motion for the operators X (t) and P(t) where the Calculate the commutator [X(t), X (0)] and show...