Consider a silver atom in a Stern-Gerlach beam splitter that distinguishes between S, = /2 and...

 

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Consider a silver atom in a Stern-Gerlach beam splitter that distinguishes between S, = /2 and S = -/2. This atom is immersed in an inhomogeneous magnetic field with B = x (@B/dx), dB/ax being constant. The atom evolves in time subject to the Hamiltonian: = KS &; x = (i) Hypothetically, if a state exists which is a common eigenstate: lel dBx m dx xx, px. (+)x)=x|x, px. (+)x) Pxlx, px. (+)x) = Px|x, px, (+)x) x | X, Px. (+)x ) = 2/ | X, Px. (+)x ), show that (px)ave is time-independent for [x, px. (+)x) that evolves in time subject to the Hamiltonian in Eq. (1). Your result in (i) would imply that there is no acceleration of atoms with S = +h/2, in contradiction with the Stern-Gerlach experiment. Henceforth, we will discard the possibility that | x. p...) exists. Instead we will work with common eigenstates of position and spin: x|x, ()x) = x|x, ()x) S|x, (+)) = //IX. (+)x). (ii) Show that the three observables (energy, S, and x) are mutually compatible, meaning every possible pair of observables is compatible. Justify why they are compatible by using [, ] = 0. (iii) Step 1 of Schrdinger's rule: determine the energy eigenstates and corresponding energy eigenvalues. Express each energy eigenstate in terms of the common eigenstates in Eq. (3). (iv) Assume the initial state is | a(0)) = |x, (-), ), which is a common eigenstate of and S,. Determine the time-evolved state | a(t)) by carrying out steps 2 and 3 of the 4-step rule. In your final expression for | a(t)), write all phase factors (multiplying |x, (+)x)) in the form etix/h, with to be determined by yourself. Consider a silver atom in a Stern-Gerlach beam splitter that distinguishes between S, = /2 and S = -/2. This atom is immersed in an inhomogeneous magnetic field with B = x (@B/dx), dB/ax being constant. The atom evolves in time subject to the Hamiltonian: = KS &; x = (i) Hypothetically, if a state exists which is a common eigenstate: lel dBx m dx xx, px. (+)x)=x|x, px. (+)x) Pxlx, px. (+)x) = Px|x, px, (+)x) x | X, Px. (+)x ) = 2/ | X, Px. (+)x ), show that (px)ave is time-independent for [x, px. (+)x) that evolves in time subject to the Hamiltonian in Eq. (1). Your result in (i) would imply that there is no acceleration of atoms with S = +h/2, in contradiction with the Stern-Gerlach experiment. Henceforth, we will discard the possibility that | x. p...) exists. Instead we will work with common eigenstates of position and spin: x|x, ()x) = x|x, ()x) S|x, (+)) = //IX. (+)x). (ii) Show that the three observables (energy, S, and x) are mutually compatible, meaning every possible pair of observables is compatible. Justify why they are compatible by using [, ] = 0. (iii) Step 1 of Schrdinger's rule: determine the energy eigenstates and corresponding energy eigenvalues. Express each energy eigenstate in terms of the common eigenstates in Eq. (3). (iv) Assume the initial state is | a(0)) = |x, (-), ), which is a common eigenstate of and S,. Determine the time-evolved state | a(t)) by carrying out steps 2 and 3 of the 4-step rule. In your final expression for | a(t)), write all phase factors (multiplying |x, (+)x)) in the form etix/h, with to be determined by yourself.

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