2 (a) Consider two observables represented by operators A and B. Show that it is possible...
(L43*) Spin can be represented by matrices. Show that all three spin matrices l 0 2 0 -1 0),"2=2 1 have eigenvalues of +1/2h and -1/2h. Calculate the corresponding eigenfunctions which we will denote as α-and β-eigenfunctions corresponding to spin l/2 particles. Show that Sj can be determined by the commutation of the other two matrices sn and sm, n, maj. Prove that the (2×2) matrix sz-s' +ss+s, commutes with all spin matrices, ie. s2s,-sis-. Calculate the eigenvalues of s2....
QM HAmiltonian, perturbation
2A where the first term is the dot product of two vectors of operators S1 and 52, W where the first term is the dot product of two vectors of operators Š and S2, which would book. The relative sign in the second term is due to the opposite electric charges of the par a) ) Find the energy eigenvalues and eigenstates of this system. V There are two states with the same energy. Suppose you perturbed...
(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...
PLEASE COMPLETE B) and stay tuned for my
following 2 questions where I will ask part c) and d). Part a) has
already been posted.
The lowest energy state of a hydrogen-like atom has total angular momentum J-1/2 (from the l-O orbital angular momentum and the electron spin s 1/2). Furthermore, the nucleus also has a spin, conventionally labeled I (for hydrogen, this is the proton spin, 1 1/2). This spin leads to an additional degeneracy. For example, in the...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E a) [10pts] What are the energy eigenvalues of the unperturbed system of the following states? 1 o 2o 0 and which energy levels are degenerate? b) [10pts Find a good basis for degenerate perturbation theory instead of c)...
1. (25 points) The Hamiltonian operator H, for a particular molecule has a complete set of orthonormal eigenfunctions on (where n = 1,2,3,...) with corresponding eigenvalues (n-1)h. The molecule in state n is subject to measurement of the dipole moment, for which the mathematical operator is represented by M. After the measurement of the dipole moment, the wave function of the particle is: Y = 0.5 0.1 +0.7071 Q. + 0.5 Pn+1 a) Show that is normalized. Now consider a...
PLEASE WRITE AS CLEAR AS
POSSIBLE
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...
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and = ħwz (6+6 +) with â (h) and at (@t) being the annihilation and creation operators for the first (second) oscillator, respectively. The Hamiltonian of two non-interacting oscillators is given by Ĥg = îl + Ħ2. Its eigenstates are tensor products of the eigenstates of single-oscillator states: Ĥm, n) = En,m|n, m), where İn, m) = \n) |m) and n, m = 0,1,2, ... a)...
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a) sin ( x) sin (y,) The corresponding energies are n) , 123 Note that the ground state, ?11 is nondegenerate with the energy E00)-E1)-' r' Now introduce the perturbation, given by the shaded region in the figure ma AH,-{Vo, if 0<x otherwise y<a/2 (a) What is the energy of the 1.st excited state of the unperturbed system? What is its degree of degeneracy,v? (b)...
part A is right above part B. Both were uploaded together
Write the four vectors S, S = 1/2,m) (see Problem 21(b)] in terms of , ,) and determine the eigenvalues. (a) J, J2, and J3 are commuting angular momentum operators. Show that the operator § = (ſ* Ì2) İ3, commutes with the total angular momentum j = 31 +32 +33. (This implies that commutes with J? as well.) (b) S1, S2, and S3 are commuting spin-1/2 operators. Let 5,...