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Quantum Mechanics II Physics 4040 Problem Set #9 To be taken up on Thursday 7 April 2005 Consider two spin ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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 ½ 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 ½ 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 "=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.