Here is instructions on how to use the simulation program.(the first simulation is used in question 5)

Transcription

1 Larmr Precessin Tutrial Here is instructins n hw t use the simulatin prgram.(the first simulatin is used in questin 5). Duble click the file sp_spins. On the tp f the new windw, click File Open Internal 3. Chse the secnd file sp_spins_abut.set 4. On the left clumn f the new windw, duble click OSP Spins Prgram 5. Duble click custm cnfiguratin 6. In the new windw, click File Lad XML, and then chse an ml file t pen the crrespnding simulatin. Initial state, measure Ŝ (Questin ~7) r. The Hamiltnian f a spin-/ particle in a magnetic field B B ˆ is Hˆ γb ˆ S, where ˆ h S is the -cmpnent f spin angular mmentum. Which f the fllwing is the eigenvalue f the Hamiltnian Hˆ γb ˆ? A. E n mγb γb B. E n m C. E h B n m γ D. E n h m S r. An electrn in a magnetic field B B ˆ is initiall in the state χ (). Which f the fllwing equatin crrectl represents the state χ (t) f the electrn? The Hamiltnian peratr is Hˆ B ˆ. γ S A. χ ( t) B. χ ( t) e iγb t / C. χ ( t) e + e iγbt / iγbt / D. χ ( t) ae + be iγbt / iγbt /

2 3. Cnsider the fllwing cnversatin between And and Carline abut the abve Hamiltnian peratr. And: Ĥ is essentiall Ŝ ecept fr sme multiplicative cnstants. Therefre, the eigenstates f Ŝ will als be the eigenstates f Ĥ. Carline: N. The presence f magnetic field will make the eigenstates f Ŝ and Ĥ different. The eigenstates f Ĥ will change with time in a nn-trivial manner. And: I disagree. If the magnetic field had a time dependence, e.g., B B cs( t) kˆ, ω the eigenstates f Ĥ will change with time in a nn-trivial manner but nt fr the present case where B r is cnstant. With whm d u agree? A. And B. Carline C. Neither 4. In the previus prblem (prblem ), suppse t satisfies γ B t / / 6. What is the prbabilit f getting when we measure at Ŝ t t? 5. Nw u can use File Lad XML larmr_-up_s_3 t check ur answer t the previus prblem (prblem 4). In the simulatin, the red blck is the magnetic field, and the blue blck is the Stern-Gerlach apparatus t measure upper detectr after SGZ clicks, the state f the particle measured is Ŝ. When the. The number 3 in the magnetic field means that the particle will sta in the magnetic field t fr time satisfing γb t / / 6 3. Eplain whether what u bserved in the simulatin is cnsistent with ur predictin in the previus questin. If it is nt cnsistent, reslve this discrepanc.

3 6. In the previus prblem (prblem ), des the prbabilit f getting time t? Eplain. depend n the Nw u can use the simulatins larmr_-up_s_3, larmr_-up_s_45 and larmr_-up_s_9 t check ur answer. In these three simulatin, the time t satisfies γb t / / 6 3, γb t / / 4 45 and γb t / / 9 respectivel. 7. Eplain whether what u bserved in the simulatin abve is cnsistent with ur predictin in the previus questin. If it is nt cnsistent, reslve this discrepanc. Initial state, measure Ŝ (Questin 8~5) r 8. An electrn in a magnetic field B B ˆ is initiall in the state χ (). After time t, the electrn will be in state χ (t). Hw can u epress χ (t) in the spin- basis and?

4 9. In the previus prblem (prblem 8), suppse t satisfies that γ B t / / 6, what is the prbabilit f getting when we measure the bservable at t t? S. Nw u can use the simulatins larmr_-up_s_3 t check ur answer. Eplain whether what u bserved in the simulatin is cnsistent with ur predictin in the previus questin. If it is nt cnsistent, reslve this discrepanc.. In the previus prblem (prblem 8), des the prbabilit f getting time t? depend n the Nw u can use the simulatins larmr_-up_s_3, larmr_-up_s_45 and larmr_-up_s_9 t check ur answer. In these three simulatin, the time t satisfies γb t / / 6 3, γb t / / 4 45 and γb t / / 9 respectivel.. Eplain whether what u bserved in the simulatin abve is cnsistent with ur predictin in the previus questin. If it is nt cnsistent, reslve this discrepanc.

5 3. Calculate the epectatin value f Ŝ at time t. Then evaluate Ŝ at time t satisfing γb t / / Find the epectatin value b cunting the number f clicks in the upper and lwer detectrs in the simulatin larmr_-up_s_3? Is this simulatin result the same as what u calculated in the previus questin (prblem 3)? 5. If u had t eplain t a friend what epectatin value phsicall means and wh the calculated value and the value frm the simulatin are the same, what wuld u sa? Initial state χ () a + b, measure (Questin 6~3) Ŝ 6. If the state f the sstem at an initial time t is given b χ () a + b, which ne f the fllwing is the state, χ (t), after a time t? The Hamiltnian peratr is Hˆ γb ˆ S. iγbt / A. χ () e ( a + b ) i B t B. e ( a + b ) γ χ () / iγbt / C. χ () e ( a + b) + ( a b) ) D. χ () ae + be iγbt / iγbt /

6 7. In prblem 6, which ne f the fllwing gives the crrect utcmes when we measure Ŝ at t? The Hamiltnian peratr is Hˆ γb Sˆ. t i / A. h / with a prbabilit ae γb t and h / with a prbabilit i Bt / B. h / with a prbabilit a e γ and h / with a prbabilit i B t / be γ b e iγbt / C. h / with a prbabilit a and h / with a prbabilit b D. h / and h / with equal prbabilit 8. Cnsider the fllwing cnversatin between And and Carline abut measuring in Ŝ the state χ (t) And: Since the prbabilit f measuring h / is a, h / is b and a + b, we can chse ur a and b as e iφ a cs( α) and b e iφ sin( α) where α, φ and φ are free parameters. Carline: I agree. Since cs ( α) + sin ( α), it gives the same relatin as a + b and there is n lss f generalit. D u agree with And and Carline? If es, fr the state χ () ( + ), what are the values f a and b when u epress it in terms f χ () a + b? And what are the values f α, φ and φ when u epress it in terms f α defined earlier as e iφ a cs( α) and b e iφ sin( α).

7 9. Suppse the initial state is χ( ) csα + sinα, which ne f the fllwing is the epectatin value Sˆ χ( t) Sˆ χ( t)? The Hamiltnian peratr is Hˆ γb ˆ. α A. cs( )cs( γbt) h / α B. sin( )sin( γbt) h / C. cs( α ) h / D. sin( α ) h / S Nw u can check ur answers f the previus prblems with a cncrete eample belw (prblem t 3). r. An electrn in a magnetic field B B ˆ is initiall in the state () ( + ) χ. Suppse t satisfies that B t / / 6. What is the prbabilit f getting γ when we measure the bservable S at t t? The Hamiltnian peratr is Hˆ γb Sˆ. Nw u can use File Lad XML larmr_-up+-dwn_s_3 t check ur answer.. Eplain whether what u bserved in the simulatin abve is cnsistent with ur predictin in the previus questin. If it is nt cnsistent, reslve this discrepanc.. In prblem, des the prbabilit f getting depend n the time t? What is the Ŝ epectatin value f?

8 Nw u can use larmr_-up+-dwn_s_3, larmr_-up+-dwn_s_45 and larmr_-up+-dwn_s_9 t check ur answer. In these three simulatin, the time t γ satisfies B t / / 6 3, B t / / 4 45 and γ γb t / / 9 respectivel. 3. Eplain whether what u bserved in the simulatin abve is cnsistent with ur predictin in the previus questin. If it is nt cnsistent, reslve this discrepanc. Initial state χ () a + b, measure (Questin 4~9) Ŝ Nw let s g back t the mst general initial state at time t which is given b χ () a + b. The Hamiltnian is Hˆ γb ˆ. Answer prblem 6 and 7. S 4. Find the state χ (t) at time t in the spin- basis and. Epress ur answer in terms f α defined earlier as e iφ a cs( α) and b e iφ cs( α). 5. Suppse the initial state is χ( ) csα + sinα, which ne f the fllwing is the epectatin value Sˆ χ( t) Sˆ χ( t)? The Hamiltnian peratr is Hˆ γb ˆ. α A. cs( )cs( γbt) h / α B. sin( )cs( γbt) h / C. cs( α ) h / D. sin( α ) h / S

9 Nw u can check ur answers f the previus prblems with a cncrete eample in which φ, φ, α / 4 (prblem 6 t 9). r 6. An electrn in a magnetic field B B ˆ is initiall in the state () ( + ) χ. The Hamiltnian peratr is Hˆ γb ˆ S. What is the prbabilit f getting at time t? Des the prbabilit f getting depend n the time t? Eplain. Nw u can use larmr_-up+-dwn_s_3, larmr_-up+-dwn_s_45 and larmr_-up+-dwn_s_9 t check ur answer. In these three simulatin, the time t γ satisfies B t / / 6 3, B t / / 4 45 and γ γb t / / 9 respectivel. 7. In prblem 6, the initial state is just χ (), but wh the prbabilit f getting is time-dependent? Can we write the state χ (t) at time t as Eplain. t e iγbt / χ ( )? 8. In prblem 6, what is the epectatin value f Ŝ at time t? Des Ŝ depend n time? When γ t satisfies that B t / / 6, what is Ŝ at t t?

10 Nw u can use larmr_-up+-dwn_s_3 t check ur answer. 9. Eplain whether what u bserved in the simulatin abve is cnsistent with ur predictin in the previus questin. Initial state χ () a + b, measure and find the general principal f time-dependence in Larmr precessin.(questin 3~39) Ŝ r 3. An electrn in a magnetic field B B ˆ is initiall in the state () ( + ) χ. The Hamiltnian peratr is Hˆ γb ˆ S. What is the prbabilit f getting if we γ measure Ŝ at t t? t satisfies that B t / / 4 Nw u can use File Lad XML larmr_-up+-dwn_s t check ur answer. 3. Eplain whether what u bserved in the simulatin abve is cnsistent with ur predictin in the previus questin. 3. Eplain in ur wn wrds wh Ŝ abve des nt depend n time where as Ŝ and Ŝ d.

11 33. Cnsider the fllwing cnversatin between Pria and Mira abut Ŝ in state χ (t) : Pria: Since the state f the sstem χ (t) evlves in time, the epectatin value Ŝ will depend n time. Mira: I disagree with the secnd part f ur statement. The time develpment f the d A i epectatin value f an peratr  is given b [ Hˆ, Aˆ ] [ ˆ, ˆ ] dt ˆ h + Aˆ t. In ur case, Sˆ H S and the peratr Ŝ des nt have an eplicit time dependence s. t d S ˆ Thus, dt With whm d u agree? and the epectatin value will nt change with time. In prblem 34~39, the Hamiltnian peratr is Hˆ B ˆ. γ 34. Chse all f the fllwing statements that are true abut the epectatin value S r in the state χ (t) : A. S r will alwas depend n time because [ Hˆ, Sˆ] B. S r cannt depend n time because the epectatin value f an bservable is its time-averaged value. S C. S r is time-independent nl when the initial state is purel r

12 35. Chse all f the fllwing statements that are true abut the epectatin value S r in the state χ (t) when the initial state is nt purel r : () The cmpnent f S r, i.e., Ŝ, is time-independent. () The and cmpnents f S r change with time. When Ŝ is maimum Ŝ is a minimum and vice versa. (3) The magnitude f the maimum values f Ŝ and Ŝ are the same. A. and B. and 3 C. and 3 D. All f the abve 36. Chse all f the fllwing statements that are true abut the epectatin value S r in the state χ (t) : () The vectr S r can be thught t be precessing abut the ais at an angle α. () The vectr S r can be thught t be precessing abut the ais with a frequencω γb. (3) All the three cmpnents f vectr S r change as it precesses abut the ais. A. and B. and 3 C. and 3 D. All f the abve

13 37. Chse a three dimensinal crdinate sstem in the spin space with the ais in the vertical directin. Draw a sketch shwing the precessin f S r abut the ais when the state f the sstem starts ut in χ () a + b makes with the ais and the precessin frequenc eplicitl.. Shw the angle that S r Shw the prjectin f S r alng the, and aes at tw separate times. Eplain in wrds wh the prjectin f S r alng the directin des nt change with time but thse alng the and directins change with time. 38. Eplain in ur wn wrds, wh fr Hˆ γb ˆ, if the initial state is Ŝ and S, Ŝ, Ŝ are all time independent. But if the initial state is ( + ) Ŝ and Ŝ depend n time while Ŝ des nt. Yur eplanatin shuld bring ut general principle f when epectatin value f an peratr will nt depend n time., then