Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .
Quantum Mechanics II
Physics 4040
Problem Set #5
To be taken up on Thursday 3 March 2005
1.
Consider a spin ½ system represented by the normalized state vector
 cosα 


 sin α eiβ 
What is the probability that a measurement of sy yields -S/2?
2-3.
A spin ½ object is in an eigenstate of sx with eigenvalue +S/2 at time t=0. At that time it
is placed in a magnetic field B = (0, 0, B) in which it is allowed to precess for a time T.
At that instant the magnetic field is very rapidly rotated in the y-direction, so that its
components are (0, B, 0). After another time interval T a measurement of sx is carried
out. What is the probability that the value S/2 will be found?
HINT: changing the B-field changes the basis. Project the initially evolved state onto the
new basis, then evolve it again. Finally project the final state onto the eigenstate of sx of
eigenvalue +S/2 and take the square.
4.
Text problem 4.27. An electron is in the spin state
 3i
χ = A 
 4
(a)
(b)
(c)
(d)
Determine the normalization constant A.
Find the expectation values of Sx, Sy, and Sz.
Find the “uncertainties”, F, for each of the three above.
Comment on which if any of the uncertainties are maximal .