Question

The Hamiltonian of an electron in magnetic field is: Hˆ = −ωSˆ z where Sˆ z is the operator of the z-component of spin. An electron has been prepared in the state |S +x 〉 at t = 0, where |S +x 〉 is the eigenstate of spin x-component with the eigenvalue of 1/2 h

(i) Evaluate the state of this electron at t = T .

(ii) Explain why this electron is said to be in a non-stationary state.

(iii) Describe the physical experiment that needs to be done in order to bring this electron to a stationary state.

Answer 1

Eigenspinors for an electron as its spin angular momentum state(one of three directions at a time) goes from are

i.) State of electron at any future time with help of time evolution operator i.e.

  When a particle is in a state which evolves through time or probability of each state is dependent on time and physical parameters e.g energy of particle (not total) changes with time i.e transitions from one state to other as time flows, describes non-stationary state. If potential of with -ve initial interacting is applied then it will return to stationary state i.e.  

And in experiment, this interacting potential(V) can be achieved by applying magnetic field along +ve z direction i.e

This field will produce interaction potential

And other option to make state stationary is to turn all magnetic field sources(i.e sources which were there already or you introduced if any) off. This ensures in either case there will no time evolution of states i.e