# 查看完整版本 : 幫忙求証物理問題?

judo123 2012-8-30 17:59

G及H 是 Hermitian operator

jmlo 2012-8-31 01:45

The following site may give you some idea how this question can be solved. (In particular the section on "Applications to Quantum Mechanics".)

[url]http://mysite.du.edu/~jcalvert/phys/groups.htm[/url]

mathfeel 2012-9-1 06:34

What does it mean for A to have certain symmetry generated by G? In term of eigenket and eigenenergy degeneracy?
If a unitary operator commutes with the Hamiltonian, what about its generator?
It's quite basic QM.

judo123 2012-9-3 09:18

1. 某物理量G = <Ĝ> = ∫ Ψ*ĜΨ dx
2. Ψ是A 的 state of a function
3. 因 物理系統A對 Generator  G 有symmetry 特性 ,所以 G= conserved, 則 dG/dt =0

Ĝ,Ĥ 及dΨ/dt 的關聯是怎樣? 思路?

jmlo 2012-9-3 13:20

As mathfeel said, you should start by thinking about the effect of the generator G on the state function of the system A. There are some special properties of the state function (recall the hints given by mathfeel in post #2) if the system is invariant under the symmetry operation dictated by the operator G (I hope my terminologies are right, mathfeel?). Using those properties, you can show that the operators G and H must commute and G is a hermitian operator.

A reference which has some examples useful to you.
Greiner and Muller, "Quantum Mechanics - Symmetries", 2nd edition, Chapter 1.

mathfeel 2012-9-3 16:52

[quote]原帖由 [i]judo123[/i] 於 2012-9-2 17:18 發表 [url=http://www.discuss.com.hk/redirect.php?goto=findpost&pid=341098652&ptid=20685204][img]http://www.discuss.com.hk/images/common/back.gif[/img][/url]

Ĝ,Ĥ 及dΨ/dt 的關聯是怎樣? 思路? [/quote]
Think in term of the Heisenberg picture. What is the relationship between dG/dt and [G, H]?

judo123 2012-9-4 10:15

ih(∂G/∂t)=[G,H],
(∂G/∂t)=0 (G= conserved),

mathfeel 2012-9-4 14:18

[img]http://latex.codecogs.com/gif.latex?1-i%5Cepsilon%20G+O%28%5Cepsilon%5E2%29[/img]

[img]http://latex.codecogs.com/png.latex?U%3De%5E%7B-iG%5Ccdot%20x%7D[/img]

ASPC 2012-9-5 11:11

judo123 2012-9-5 18:52

ASPC 2012-9-6 09:50

ih(∂G/∂t)=[G,H],
(∂G/∂t)=0 (G= conserved),

ih(∂G/∂t)=[G,H], Textbook會問呢條eq嘅証明方法,証明後再得出甚麽結論(結論就係G是conservative quantity嘅條件喺  [G,H]=0) 你冇可能用結論來証明結論,條題目問因G=conserved,証明[G,H]=0,

Ψ是A 的 state of a function (wave function postulates)
symmerty operations generated by G ,所以 G= conserved, 則 dG/dt =0 (條題嘅意思)
dG/dt =d <Ĝ>/dt =∫{( dΨ*/dt)ĜΨ+Ψ*(dĜ/dt)Ψ +Ψ*Ĝ(dΨ/dt)dx
ih(∂Ψ/∂t)=ĤΨ or -ih(∂Ψ*/∂t)=Ψ*Ĥ+
dG/dt =i/h∫{(ĤΨ)*ĜΨ-Ψ*Ĝ(ĤΨ)}dx     把dΨ/dt and dΨ*/dt 攪定!!
dG/dt=i/h∫{Ψ*(ĤĜ-ĜĤ)Ψ}dx    (ĤĜ-ĜĤ)得出呢個關係,以後點做,你識卦!!!

judo123 2012-9-7 10:34

Thanks!

lightspeed2020 2022-2-10 08:40

[quote]原帖由 [i]ASPC[/i] 於 2012-9-6 09:50 AM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=341399335&ptid=20685204][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

ih(∂G/∂t)=[ ... [/quote]

rhwlam 2022-4-9 09:19

topochu 2022-4-10 12:40

[url]https://www.damtp.cam.ac.uk/user/dbs26/PQM/chap4.pdf[/url]

rhwlam 2022-4-18 13:35

topochu 2022-4-18 16:46

[quote]原帖由 [i]rhwlam[/i] 於 2022-4-18 13:35 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=547684256&ptid=20685204][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

[attach]13217263[/attach]

topochu 2022-4-18 17:35

[attach]13217320[/attach]

topochu 2022-4-18 17:52

Redundant

[[i] 本帖最後由 topochu 於 2022-4-18 21:12 編輯 [/i]]

topochu 2022-4-18 17:54

[[i] 本帖最後由 topochu 於 2022-4-18 21:13 編輯 [/i]]

rhwlam 2022-4-18 19:36

[quote]原帖由 [i]topochu[/i] 於 2022-4-18 17:52 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=547690724&ptid=20685204][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

(當中存在問題. 為免誤導, 已刪除.)

[[i] 本帖最後由 rhwlam 於 2022-4-19 07:47 編輯 [/i]]

rhwlam 2022-4-18 19:49

topochu 2022-4-18 21:25

[quote]原帖由 [i]rhwlam[/i] 於 2022-4-18 19:36 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=547694074&ptid=20685204][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

13217681 [/quote]
「因為G係symmetry operation嘅generator，所以dG/dt=0」係一個結果而唔係前提，所以第一步有問題。同埋透過Heisenberg equation of motion，下一步已經可以得出[H,G]=0。同埋點解第二步會假設∂G/∂t。

[[i] 本帖最後由 topochu 於 2022-4-18 21:32 編輯 [/i]]

rhwlam 2022-4-18 22:21

[quote]原帖由 [i]topochu[/i] 於 2022-4-18 21:25 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=547697775&ptid=20685204][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

「因為G係symmetry operation嘅generator，所以dG/dt=0」係一個結果而唔係前提，所以第一步有問題。同埋透過Heisenberg equation of motion，下一步已經可以得出[H,G]=0。同埋點解第二步會假設∂G/∂t。 [/quote]
——

[[i] 本帖最後由 rhwlam 於 2022-4-18 22:49 編輯 [/i]]

topochu 2022-4-18 22:53

[quote]原帖由 [i]rhwlam[/i] 於 2022-4-18 22:21 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=547699845&ptid=20685204][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

dG/dt是基於noether’s theorem. [/quote]
Noether (first) theorem只係話每一個generator of a continuous symmetry都會有一個對應嘅conserved quantity。好似今次個case, generator G會對應一個conserved quantity Q（通常都唔係G，比如話U(1) gauge symmetry個對應嘅conserved charge就[i]可以[/i]係electrical charge），而呢個Q係可以再用variation of Lagrangian推導出嚟。

[[i] 本帖最後由 topochu 於 2022-4-18 23:18 編輯 [/i]]

topochu 2022-4-18 23:11

[attach]13218318[/attach]

rhwlam 2022-4-19 05:42

[quote]原帖由 [i]topochu[/i] 於 2022-4-18 22:53 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=547700934&ptid=20685204][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

Noether (first) theorem只係話每一個generator of a continuous symmetry都會有一個對應嘅conserved quantity。好似今次個case, generator G會對應一個conserved quantity Q（通常都唔係G，比如話U(1) gauge symmetry個對應嘅conserved charge就可以係electri ... [/quote]

topochu 2022-4-19 07:51

topochu 2022-4-20 16:44

[quote]原帖由 [i]rhwlam[/i] 於 2022-4-18 22:21 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=547699845&ptid=20685204][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

—— [/quote]

[attach]13222557[/attach]

[[i] 本帖最後由 topochu 於 2022-4-20 16:46 編輯 [/i]]

rhwlam 2022-4-20 21:56

[quote]原帖由 [i]topochu[/i] 於 2022-4-20 16:44 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=547750606&ptid=20685204][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

13222557 [/quote]