Consider the observable q_u = u_1q_1+u_2q_2+u_3q_3 ,

where the q_i are the Pauli matrices and

where u = (u_1,u_2,u_3) in R^3

and the Hamiltonian is H = q3.

Compute the observable (q_u)_H (t) derived from q_u in the Heisenberg picture.

Solution Preview

The Heisenberg picture is obtained as follows:

A_H = Exp[i H t/h-bar] A_S Exp[-i H t/h-bar]

Here A_S is the operator in the Schrodinger picture and H is the Hamiltonian. The Hamiltonian is given by sigma_z. We thus need to exponentiate sigma_z. This is not difficult, because sigma_z is already diagonal in the standard basis. I.m.o., this problem is too easy because of this. So, after you do this problem you should generalize the Hamiltonian. Instead of sigma_z you take:

H = h_x sigma_x + h_y sigma_y + h_z sigma_z = h dot sigma

Your problem corresponds to the special case: h_x = h_y = 0 and h_z = 1.

You now need to exponentiate this matrix. If we define |h| as the norm of the vector (h_x,h_y,h_z), then we have:

Suppose that particle 1 (with spin 1) and particle 2 (with spin 1/2) interact via the Hamiltonian operator
H = lambda S1 dot S2 , where lambda is a constant.
Calculate the energy of the state |s,ms> .

A. Consider a system of 2 particles: particle 1 has spin 1, and particle 2 has spin 1/2. Let S be the total angular momentum operator of the two particles, where the eigenvalues of S^2 and Sz are ħ^2s(s+1) and ħms, respectively. The particles are in the state s= 3/2 and ms = 1/2.
Calculate the wave function |s = 3/2

QUESTION 16a:
Consider a 2470-lb automobile clocked by law-enforcement radar at a speed of 85.5 pmh (miles/hour). If the position of the car is known to be within 5.0 feet at the time of the measurement, what is the uncertainty in the velocity of the car?
QUESTION 16b:
If the speed limit is 75 mph, could the driver of th

See attached file for equations and answer the following:
(a) Calculate the matrix representing in the { | + >, | - > } basis, the operator H, the Hamiltonian of the system.
(b) Calculate the eigenvalues and the eigenvalues of H
(c) The system at the time t = 0 is in the state | _ >. What values can be found if the energy

(a) Let Q be an operator which is not a function of time, and let H be the Hamiltonian operator. Provide proof for an equation (see attached file for equation).
Here {q} is the expectation value of Q for an arbitrary time-dependent wae function , which is not necessarily
an eigenfunction of H, and {[Q,H]} is the expectatio

We have to use the overload operatoroperator == to compare to objects and we are to use operator !=. The rest of the program does as I ask it to. I will go ahead and the instructions given to us anyway to help better understand the question. I will also post my .cpp and .h files. I am writing the data to an outfile. Also, it M

Can you show that the energy of interaction is proportional to the scalar product s∙l.? See attachment for symbols.
The energy of a magnetic moment mu in a magnetic field B is equal to their scalar product (see attachment). If the magnetic field arises from the orbital angular momentum of the electron, it is proportional to

What is the approximate uncertainty in the velocity of a proton known to remain within a nucleus of diameter 2.8 * 10^-15 m? What kinetic energy would the proton (mass approximately 1.6 * 10^-27 kg) have at this velocity?

A spin 1/2 particle is in the state |Psi> = Sqrt[2/3] | Up > + Sqrt[1/3] | Down >
Suppose a measurement is made of thespin in the z direction and the result is m_s = -1/2. Now a second measurement is made to determine thespin in the x - direction. What is the probability thespin will be in the +x direction?
So I underst