میدانیم در ترمودینامیک کلاسیک در فرایند آدیاباتیک احتمالات حالات ثابت میمانند. این مسأله به مکانیک کوانتومی هم تعمیم داده- میشود؛ به این معنی که اگر حالت سازست خالص باشد و به صورت یک بسط بیان شود ضرایب بسط در فرایند ثابت هستند و کتهای پایه تغییر میکنند. اگر سازست در یک حالت مخلوط باشد شرط آدیاباتیک بودن فرایند ثابت بودن ماتریس چگالی است. آیا میتوانیم یک فرایند آدیاباتیک داشته باشیم که در آن عملگر همیلتونی چرخهای باشد، یعنی جملهی پتانسیل وابسته به زمان آن در آغاز و پایان فرایند برابر صفر شود؟
Quantum Mechanics, Thermodynamics, Information and Complexity
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4 comments:
کیوان عزیز من خوب متوجه نشدم! چرا در فرایند بی دررو حالات ترمودینامیکی باید ثابت بماند؟
ا. ش.
It is possible. Suppose that the change of potential goes along a cyclic curve as the following: it starts from zero and increase linearly up to the time T, and afterward, it decreases linearly with the same rate as it increased. Then at the time 2T the potential becomes zero again. The unitary operator describing the transformation of the state in the period [0,T] is the adjoint of the unitary operator describing the transformation of the state in the period [T,2T]. Accordingly, they nullify each other and the corresponding state remains unchanged.
By the way, my reasoning is indifference to the state of the system: to be mixed or pure, the result is the same!
Keyane aziz in post ro didam va be mahze khondane on nokate ziyadi be nazaram resid ke bazi az onha ro inja kholase mikonam. Etminan daram ke shoma va baghiye dostan in matalleb ro bayad bedonan. Gaman mikonam shoma in matlab ro baraye bahs bishtar inja neveshtid.
Man say khaham kard kenare dostanam ghalam bezanam ina. Bessiyar hatake zibaeest ke shoma aghaz kardid. Roye man ham be onvane ye bandeye kamtarin oo yare hamrah mitonid hessab koni. Man az hamrahiye Dr. Aziz dar bode keihani bessiyar amokhtam va hamishe in soal barayam bood ke chera weblogi almi nist be rah nemiadazan azizam. Az didane inja kheili khosh-hal shodam
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A comment on conceptual foundation of quantum thermodynamics in the above post:
As far as, I understand what you argued MUST be related more explicitly to the time evolution of the state vectors in Heisenberg picture (neglecting the possibility that it might be also related to more complicated interaction picture too!!?). Because “basis” vectors in Schrodinger picture must be stationary. By the way, I try to dig out some facts here:
1. There is not any correspondence for quantum state in classical words. For any comparison purposes, one SHOULD compare the operators in quantum mechanics with classical observables. Not states! Moreover, Heisenberg Pictures in which Operators are time dependent can be used for the comparison more intuitively or naturally.
2. I somehow believe: Adiabaticity assumption in thermodynamics is QUITE DIFFERENT from adiabatic approximation in quantum mechanics (in standard speaking albeit). In thermodynamics adiabaticity means, no change in the enthalpy of the system, but does not mean the state of a system remain unchanged, never! For instance, surely it would change when pressure or volume of the system is in change. On the other hand, in quantum(Q) dynamics adiabaticity is an approximation for describing dynamics of a slowly evolving Q system from system i to system j (surely Hamiltonian of the system would change) in such a way that state of the system have to stay in same state.( i.e. from nth state to the nth state of the new system). This is adiabatic approximation in quantum mechanics in standard definition. I would be happy if someone makes it clear how I confused it!!
3. On the basis of my understanding, I strongly believe the problem is much more easier than what has been stated at this post in all respect. If we see it in a simple way, we need just simple basics to proof or understand that. In quantum mechanics within the Hilbert space for any state we would have a state operator, correspondingly. This operator is actually a projector, i.e it has just two eigenvalues of 1 or 0 (by definition of projector). The physical description is simply whether we have or don’t have a particle in the system! This operator is known as density matrix for an arbitrary state in the Hilbert space!! (Also recall the first postulate in quantum mechanics that any state in Hilbert space can be represented on the basis of eigenvectors of any observables!!!). The case that the arbitrary state is an eigenvector (pure state), is the simplest case of course. Then mix state can be described by the density matrix which gives us the probabilities of all states (of course the sum-trace- must be one or zero). OK, very simplely till now the physics is whether or not there is any particle in a system and nothing more. Quantum evolution means action of the unitary transformation on a ket. Of course the norm would stay conserve under such a transformation by definition!! In order to test our understanding who can say? In what conditions the norm would not stay conserve?? The answer is very simple. (Hint: there are two conditions in which norm would not stay conserve during time evolution. Assume the state is normalized at first. Using the first equation in quantum mechanics, i.e time-dependent Schrödinger equation!).
4. After all, what do you mean by probabilities of the state in classical thermodynamics!!! I got it, you need to connect the meaning of ensemble and density matrix to the classical adiabatic system anyhow! At least it need more clarification. The best comparison and discussion must be based on the trajectories in phase spaces and setting up the meaning of ensemble for two case. Then compare the space-momentum-time evolution with in the two frame works (classical and quantum version). This is well-defined conncetion between the two formalism and approach.
Best wishes;
Soshiyans.
PS: all comments in the comment is appreciated in advance. :)
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