3. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the...
Rotational states of a diatomic molecule can be approximated by those of a rigid rotor. The hamiltonian of a rigid rotor is given by hrotor 12/21, where L2 is the operator for square of angular momentum and I the moment of inertia. The eigenvalues and eigenfunctions of L2 are known: Lylnu =t(1+1)ay," , where m.--1, , +1 a) Calculate the canonical partition function : of a rigid rotor. Hint: Replace summation over by integral. b) What is the probability that...
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the subscript of E is n, the principal quantum number. The other two numbers are the 1 and m values, find the expectation values of H (you may use the eigenvalue equation to evaluate for H), L-(total angular momentum operator square), Lz (the z-component of the angular momentum operator) and P (parity operator). Draw schematic pictures of 1 and...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
A plane rotator is a system whose wave functions and energy levels are h2m2 21 I is the moment of inertia of the rotator. A perturbation is introduced of the form where λ is a small parameter and k is a non-zero integer. (a) What are the adapted unperturbed wavefunctions? (b) Calculate the energy levels of the perturbed system to O(A2). A plane rotator is a system whose wave functions and energy levels are h2m2 21 I is the moment...
3. (18 points) The angular momentum operator in the y direction is given by: ly- while the position operator in the x direction is given by: & x. a. (10 points) Determine the commutator for these operators when applied to the dummy function f(x). b. (8 points) What does the value of the commutator tell us about the relationship between the quantum mechanical observables associated with these two operators? Explain 3. (18 points) The angular momentum operator in the y...
2.The angular momentum is L = p a) What is the representation of the angular momentum operator b)Use the polar coordinates to compute L o)Show that the eigenfunction forp) m(p),where mis integer. What is the Τηφ(p),where What mis integer. is the eigenfunction φ(p), assume 0 (p) 2π 2.The angular momentum is L = p a) What is the representation of the angular momentum operator b)Use the polar coordinates to compute L o)Show that the eigenfunction forp) m(p),where mis integer. What...
3. (a) Draw a vector model figure illustrating the orbital angular momentum L and orbital magnetic dipole moment μ vectors for a typical atomic state, their z- components, and their Larmor precession in a uniform magnetic field B for an electron in the p-state of hydrogen. orbital magnetic dipole moment vectors, their sum the total magnetic dipole mo- b) Draw vector model fgures indicating the total angular momentum posibilities and (c) For one of the above states, draw a vector...
Calculate the classical frequency of rotation of a rigid body whose energy is given by Eq. (8.9) for states of J =J and J =J + 1, and show that the frequency of the spectral line associated with a transition between these states is intermediate between the rotational frequencies of the states. 14. Calculate the classical frequency of rotation of a rigid body whose energy is given by Eq. (8.9) for states of J = J and J J+1, and...
1. (25 points) The Hamiltonian operator Ho for a particular molecule has a complete set of ions фп (where n-1,2,3, ) with corresponding eigen values (n2-1)r. Ho The molecule in state n is subject wave fial operator is to measurement of the dipole moment, for which the n of the rpresented by M. After the measurement of the dipole moment, the measu a) Show that W is normalized. er a system with a Hamiltonian which can be written: Here c...
Problem Set VII (Quantum Physics) 1. The Hamiltonian for an axially symmetric rotator is 213 where Ii and Is are the moments of inertin. (a) What are the eigenvalues of H? (Ans: where l 2m 2 -1.) (b) What is the spectrum in the limit when I is much larger than 1,? (Ans: E ~ (n°/213)㎡.) Problem Set VII (Quantum Physics) 1. The Hamiltonian for an axially symmetric rotator is 213 where Ii and Is are the moments of inertin....