Homework Help is Here – Start Your Trial Now!
Science
Advanced Physics
The spin of an electron is described by a vector = and the spin operator S = S,i Šj + S,k with components S, -(1:). =(7) . -( 1 0 -i %3D 1 0 0 -1 (a) (i) State the normalisation condition for . (ii) Give the general expressions for the probabilities to find Sz = th/2 in a measur ment of S. (iii) Give the general expression of the expectation value (S:).

The spin of an electron is described by a vector = and the spin operator S = S,i Šj + S,k with components S, -(1:). =(7) . -( 1 0 -i %3D 1 0 0 -1 (a) (i) State the normalisation condition for . (ii) Give the general expressions for the probabilities to find Sz = th/2 in a measur ment of S. (iii) Give the general expression of the expectation value (S:).

icon
Related questions
Question

The spin of an electron is described by a vector [psi] = mat([psi_up],[psi_down]) and the spin operator S = Sxi+Syj+Szk with components Sx = (h/2)*mat([0,1],[1,0]), Sy = (h/2)*mat([0,-i],[i,0]), Sz = (h/2)*mat([1,0],[0,-1]).

a)i) State the normalisation condition for [psi].

ii) Give the general expressions for the probabilities to find Sz =+-(h/2) in a measurement of Sz.

iii) Give the general expression of the expectation value <Sz>.

b)i) Calculate the commutator [Sy,Sz]. State whether Sy and Sz are simultaneous observables. 

ii) Calculate the commutator [Sx,S2], where S2 = Sx2 + Sy2 + Sz2. State whether Sx and S2 are simultaneous observables. 

c)i) Show that state [phi] = (1/sqrt(2))*mat([1],[1]) is a normalised eigenstate of Sx and determine the associated eigenvalue.

ii) Calculate the probability to find this eigenvalue in a measurement of Sx, provided the system is in the state [phi] = (1/5)*mat([4],[3]).

iii) Calculate the expectation values <Sx>, <Sy>, <Sz> in the state [psi].

Image with better formatted question is attached!

The spin of an electron is described by a vector p: and the spin operator S = S,i+ (? ). 0 -i 0 1 1 0 1 0. Sj+ S,k with components S, Sy i 1 (a) (i) State the normalisation condition for v. (ii) Give the general expressions for the probabilities to find Sz = +h/2 in a measure- ment of S. (iii) Give the general expression of the expectation value (S.). (b) (i) Calculate the commutator [Šy, Š]. State whether S, and S, are simultaneous ob- %3D servables. (ii) Calculate the commutator [S„, Š³], where S = S? + S? + S?. State whether S, and S° are simultaneous observables. (c) (i) Show that the state o V2 is a normalised eigenstate of S, and determine the associated eigenvalue. (ii) Calculate the probability to find this eigenvalue in a measurement of S, provided 1(4) 53 the system is in the state (iii) Calculate the expectation values (S), (Š,) and (S;) in the state y.
Transcribed Image Text:The spin of an electron is described by a vector p: and the spin operator S = S,i+ (? ). 0 -i 0 1 1 0 1 0. Sj+ S,k with components S, Sy i 1 (a) (i) State the normalisation condition for v. (ii) Give the general expressions for the probabilities to find Sz = +h/2 in a measure- ment of S. (iii) Give the general expression of the expectation value (S.). (b) (i) Calculate the commutator [Šy, Š]. State whether S, and S, are simultaneous ob- %3D servables. (ii) Calculate the commutator [S„, Š³], where S = S? + S? + S?. State whether S, and S° are simultaneous observables. (c) (i) Show that the state o V2 is a normalised eigenstate of S, and determine the associated eigenvalue. (ii) Calculate the probability to find this eigenvalue in a measurement of S, provided 1(4) 53 the system is in the state (iii) Calculate the expectation values (S), (Š,) and (S;) in the state y.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer