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rar0308
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E is finite and V is infinite, So T should be negative infinite
DrClaude said:Can you please give more details about the situation you are contemplating? "delta function wall" is not very clear.
rar0308 said:Yes, what stevendaryl said is exactly what I'm curious about.
dauto said:Yes, Kinetic energy can be negative in quantum mechanics which leads to penetration into a region that would be forbiden in classical mechanics allowing for instance for tunnel effect.
Jazzdude said:No. Kinetic energy is not a position dependent property. To be precise, the only sensible definition of kinetic energy is the quadratic form <psi|T|psi> for normalized |psi>. And such a state |psi> which is entirely contained within the region where you'd classically expect "negative kinetic energy" has an localization induced momentum uncertainty that makes the total kinetic energy positive.
There's a theorem which is not hard to prove that <psi|T|psi> is always >0 for states on the hilbert space of square integrable functions, because otherwise they could not be normalized.
Cheers,
Jazz
dauto said:We're not talking about exactly the same thing. The expected value will be positive. But the question is about the value of the function before the integration is performed to find the expected value. At least that's how a read it.
Jazzdude said:What function? There is no kinetic energy function is quantum theory. And the expectation value IS the kinetic energy.
Cheers,
Jazz
stevendaryl said:Well, I would say that kinetic energy is an operator, not an expectation value.
For any operator [itex]\hat{O}[/itex] and any wave function [itex]\Psi(x)[/itex], we can define a position-dependent (and time-dependent, if [itex]\Psi[/itex] is time-dependent) function [itex]O_Q(x)[/itex] via:
[itex]O_Q(x) = \dfrac{\hat{O} \Psi(x)}{\Psi(x)}[/itex]
If [itex]\Psi(x)[/itex] is an eigenstate of [itex]\hat{O}[/itex] then [itex]O_Q(x)[/itex] is the eigenvalue. Otherwise, [itex]O(x)[/itex] is in general a complex-valued function.
Defining such functions can be useful in seeing how QM approaches the classical limit. For example, for the special case of momentum, we have:
[itex]p_Q(x) = -i \hbar \frac{d}{dx} ln(\Psi(x))[/itex]
The corresponding classical "momentum function" is:
[itex]p_C(x) = \dfrac{\sqrt{2m (E - V(x))}}{\hbar}[/itex]
Jazzdude said:Kinetic energy is a property of a state.
That's only a sensible construction in the context of approximations (like WKB), but not as part of the structure of quantum theory itself. Specifically you cannot formulate conservation laws or frame changes coherently with such a function.
stevendaryl said:I don't think that makes any sense, any more than saying that the position is a property of the state. In non-relativistic quantum mechanics, the position operator has the same sort of status as the kinetic energy.
My point is that there is a way to make sense of a position-dependent kinetic energy (or momentum, or whatever). It's useful for understanding the relationship between quantum mechanics and the classical limit.
Jazzdude said:Kinetic energy cannot be state independent (like being just the operator you suggested), instead, it must be a function of the state of the system. We know that kinetic energy is a sensible concept, even in quantum mechanics, because of the conservations laws that involve it. That makes it a property of the state. It doesn't mean that a state necessarily has to be in a kinetic energy eigenstate. It's the "expectation value" (which is really a misnomer, because it automatically implies you perform a measurement) for which the conservation law holds, so that is the proper definition of kinetic energy in a quantum context.
And my point is that this is highly misleading. Suggesting that there is any such property as kinetic energy that you can associate with position implies an underlying structure of some sort where the particle is actually located somewhere, in form of a single point, but we just don't know where.
This is exactly the kind of comparison to classical physics that confuses the heck out of students and results in weird debates about the real meaning of quantum physics.
It also implies a much too simplified and superficial transition from quantum to classical. But most importantly, there is no way to make sense of any such definition in terms of conservation laws.
stevendaryl said:I don't understand what you're saying. Why is kinetic energy more of a "property of state" than position is?
How does having a position-dependent kinetic energy function imply such a thing, any more than having a position-dependent wave function [itex]\Psi(x)[/itex]?
I think avoiding confusing students is way over-rated. Confusion is an important stage in reaching understanding. I think for any given topic, such as quantum mechanics or relativity, or whatever, the more different ways you know how to think about it, the better.
What's the proof that there is no way to "make sense of any such definition in terms of conservation laws"?
Actually, understanding the classical limit using "momentum functions" was the topic of research done by my advisor years ago. I thought it was very interesting, even though not much came of it.
Jazzdude said:There is a conservation law for energy(-momentum), and that law holds independently from the system being in a corresponding eigenstate.
But this is not a good or meaningful way to think about it, so it doesn't really help you to get a different perspective on things.
Anyone can research whatever he wants. But whatever comes out, it will certainly not revolutionize our understanding of energy conservation in quantum theory or the quantum to classical transition.
stevendaryl said:But kinetic energy by itself isn't conserved, so I'm not sure how it's relevant that total energy is conserved. They are two different operators.
It doesn't make any sense to me for you to say that the kinetic energy is a function of state, and the potential is not. But the potential is explicitly position-dependent, so it can't be a function of state.
Well, I think that what you are saying is not a good or meaningful way to think about it. So there.
You're making claims that are beyond what you actually know.
On the other hand, I DO know what I'm talking about, and I agree that nothing much came of Dr. Leacock's research. So you're right, but a guess doesn't count for much.
Jazzdude said:Going from total energy to kinetic energy is only a matter of fixing a frame, which I assumed to have happened.
Now that's a claim that is beyond your knowledge.
So what are you arguing about?
stevendaryl said:I'm sorry, that doesn't make any sense. How does "fixing a frame" convert kinetic energy into total energy, or vice-versa? They are different things.
What you are saying doesn't seem to make any sense.
Because you are saying incorrect things to argue about something that you don't know anything about. You don't know anything about Hamilton-Jacobi theory, but you are making pronouncements about how meaningless it is.
Jazzdude said:If that doesn't make sense to you then you might consider to refresh your knowledge on relativistic energy and the Poincaré group, not even specifically in quantum theory.
And claiming I don't understand Hamilton-Jacobi (which is almost entirely unrelated to what we're discussing here) is not really helpful.
stevendaryl said:Let me be a little more blunt: what you're saying is WRONG. We're not talking about relativistic quantum mechanics here, we're talking about nonrelativistic quantum mechanics. So that claim is at best a non-sequitur. But I don't see how it makes any sense from the point of view of relativistic quantum mechanics, either.
stevendaryl said:Except that the quantum Hamilton-Jacobi equation is exactly what I am talking about. So it doesn't make a bit of sense for you to say that it is almost entirely unrelated.
Jazzdude said:So then, what exactly is wrong, and how?
You really stopped discussing in this thread. I gave arguments and tried to explain the issues as good as I can.
Jazzdude said:But it is not what I am arguing about. I was always referring to your statements long before you pointed to HJ. And what I said never applied to that either.
stevendaryl said:I have pointed it out several times. Let's take an example of the harmonic oscillator: The Hamiltonian is:
[itex]H = -\frac{\hbar^2}{2m}\frac{d}{dx^2} + \frac{1}{2}k x^2[/itex]
The first term on the left is the kinetic energy operator, and the second term is the potential energy operator. The sum of the two is the total energy operator.
You said that "Going from total energy to kinetic energy is only a matter of fixing a frame."
That makes no sense whatsoever.
You said words. I wouldn't say you actually gave any arguments that made any sense.
Jazzdude said:No, you replied to my argument of why kinetic energy is a sensible physical property of a state, and I was never referring to the harmonic oscillator.
stevendaryl said:But the whole discussion has been about kinetic energy in the context of a potential function. When there's a potential function, the kinetic energy is not conserved. This is pretty basic stuff, and you seemed to be disputing it.
A delta function wall is a theoretical construct in physics that represents an infinitely thin and infinitely high potential barrier. It is often used in quantum mechanics to model the behavior of particles in confined spaces.
Yes, a particle can have negative kinetic energy in a delta function wall. This is because the delta function wall creates a potential energy barrier that can cause the particle to have a negative kinetic energy, meaning it is moving in the opposite direction of the barrier's force.
A particle's kinetic energy can change in a delta function wall due to the potential energy barrier. When the particle approaches the barrier, its kinetic energy decreases as it is repelled by the barrier's force. If the particle passes through the barrier, its kinetic energy may become negative.
Yes, negative kinetic energy in a delta function wall is physically possible in the realm of quantum mechanics. However, it is not observed in classical mechanics, as it violates the principle of conservation of energy.
The presence of negative kinetic energy in a delta function wall can significantly impact the behavior of a particle. It can cause the particle to reflect off the barrier, tunnel through it, or become trapped in a bound state. These effects are essential in understanding the behavior of particles in confined spaces and in the development of quantum mechanical models.