The Hamiltonian is given by: a and b are real and constant 1). Find the allowed...
Consider the Hamiltonian, , where and
a) Determine the energies and eigenfunctions of the
undisturbed Hamiltonian.
b) Find the corrections to the energies using
perturbation theory of first order, as well as their corresponding
wave functions.
In a given representation, the matrix representing the
Hamiltonian of a particle is given by
with 0 < ε < 1. Find the energy eigenvalues and
eigenfunctions of the particle in the representation.
0002 120 000-1 02 -1000 -2000 2 0
The Hamiltonian of a system in the basis In > is given by H = hw(" >< 0,1 + il" >< 421-142 >< 0,1 -21°3 >< $3D Here w is a constant. Write the Hamiltonian in the form of a matrix and obtain its eigenvalues and eigenfunctions. Express the eigenfunctions in terms of the basis In > and in its eigenvalues as En = hwe If the system is initially in the state | (0) >= 10 > a. What...
Q10 The Hamiltonian of a two-state system is given by H E ( i)- I02)(2 | -i | ¢1)(2 | +i | ¢2) (¢1 1) where , p2) form a complete and orthonormal basis; E is a real constant having the dimensions of energy (a) Is H Hermitian? Calculate the trace of H (b) Find the matrix representing H in the | øı), | 42) basis and calculate the eigenvalues and the eigenvectors of the matrix. Calculate the trace of...
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5. A quantum system is described by the one-dimensional Hamiltonian (in units here 1) d2 dz2 Notice that this Hamiltonian has the potential energy of x2 (we will soon see that this Hamil tonian describes a good model of molecular vibration). Let us consider the two wavefunctions (a) Show that h(z) and 2(z) are eigenfunctions of this Hamiltonian and find their corre- sponding eigenvalues. (b) Find the constants Ai and A2 that normalize the...
The hamiltonian of a perturbed system it's given by:
a ) Find
b ) Diagonalize the Hamiltonian
, find
(for the complete Hamiltonian) and expand to second order
c ) Find
and
and compere with subsection (b)
Г/1000) /λ 0 0 0 0 8 0 0 й-Е01 0 0 3 0 0 0 _2λ 0 0 0 7
(introduction to quantum mechanics)
, the Hamiltonian matrix is H- 3. In the basis |1) - (a) Find the eigenvalues En and eigenfunctions Ion) of H. (b) The system is in state 2) initially (t 0). Find the state of the system at t in the basis n). (c) Calculate the expectation value of H. Briefly explain your result. Does it depend on time? Why?
, the Hamiltonian matrix is H- 3. In the basis |1) - (a) Find the...
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,...
(3)Consider an atomic p-electron (-1) which is governed by the Hamiltonian H-Ho +Hl,where Ho=a L,.bhand H,-./2 where a,bandcare nonzero real numbers with a 굶b. (a) Determine the Hamiltonian in Matrix form for a basis | I,m > with 1-land ,n = 0,±1. You may use the formula (b)Treat H,as a perturbation of Ho. What are the energy eigenvalues and eigenfunctions of the unperturbed problem? (c)Assume as>lcl and bsslcl. Use perturbation theory to calculate eigenvalues of H to first non trivial...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...