# cos sin α α β ei P t V t = − − ( ) sin ( ) / ( ) 4 2 ω ω ω ω P P t t

advertisement ```Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
Quantum Mechanics II
Physics 4040
Problem Set #9
To be taken up on Thursday 7 April 2005
Consider two spin &frac12; particles, whose spins are described by the Pauli operators F1 and F2.
Let n be the unit vector connecting the two particles and define the operator
1.
S12 = 3( [email protected] n)([email protected] n) - F1 @ F2
Show that if the two particles are in an S=0 state (singlet) then
S12 Xsinglet = 0
Show that for a triplet state
(S12 - 2)(S12 + 4) Xtriplet = 0
(Hint: Choose n along the z-axis)
2.
Consider a spin &frac12; system represented by the normalized state vector
 cosα 


 sin α eiβ 
(a)
(b)
(c)
3.
Find the expectation value of Sx.
What is the probability that a measurement of Sy yields -S/2?
For &quot;=B/4 and \$=B/4, what is the porbability that a measurement of Sy yields S/2?
We were working with the expression for the probability as a function of time for a
transition to occur between two states given by eqtn 25 in your notes:
2
Pa → b (t ) =
4 Vab sin 2 (ω 0 − ω )t / 2
h 2 (ω 0 − ω ) 2
Examine the behaviour of this expression as a function of T for three different time
values; i.e. plot the function vs T for t=1/T0, 10/T0, and 100/T0.
4.
Show that with PaPb(t) as given above that
Pa → b ≡ lim
t→∞
Pa → b (t ) 2π
2
= 2 Vab δ (ω 0 − ω )
t
h
With the definition of the delta function given in your notes.
```