Thanks for the reply. Let me clarify my question. When we measure we only get the eigenvalue, but we have said that the system was in SUPERPOSITION of many values, now for position for instants maybe I don't see a serious ramification but for energy its a bit hard for me to understand in regards to conservation of energy.Our measuring device has to interact with the particle to measure its energy. That interation involves an exchange of energy, and the total energy of the quantum system consisting of the particle and the device is conserved.
Thanks for the detail reply. Just like a good student most of the time I don't ask too many questions just do the calculations. But once in a while some questions pop into my head (usually not explicitly stated in textbooks) and upon searching for the answer I am relieved that I am not alone.I think you have a serious misconception about QT.
Based on Nugatory's reply, the short answer is, in your terms, the excess energy goes to the measuring system.It is said that for a particle in a box the energy is in a superposition. If indeed that is the case what happens when a measurement is made where does the excess energy go. Of course, that is based on my understanding is that superposition is a real physical and not platonic.
Suppose the particle is in a superposition of energy E1 and E2. If the measuring device measures the energy of the particle to be E1, the device will have, after measurement, energy E3. If the measuring device measures E2, the device will have energy E4. And E1+E3=E2+E4, so energy is conserved.Our measuring device has to interact with the particle to measure its energy. That interation involves an exchange of energy, and the total energy of the quantum system consisting of the particle and the device is conserved.
When the system is in a superposition of two or more energies, one should think of it as a system that does not have a well defined energy at all. A measurement of energy then gives energy a well defined value. Violation of energy conservation would occur if the system evolved from a state with one value of energy to a state with another value of energy. But here it is not what happens, instead we have an evolution from a state with no value of energy to a state with a value of energy. Hence we cannot say that conservation of energy is violated.It is said that for a particle in a box the energy is in a superposition. If indeed that is the case what happens when a measurement is made where does the excess energy go. Of course, that is based on my understanding is that superposition is a real physical and not platonic.
Does the conservation of energy imply that whenever a particle A is created to be in a superposition of energy ##E_1## and ##E_2##, another particle B must also be created in a superposition of energy ##E_1+c## and ##E_2+c## such that if A is measured to have energy ##E_1##, B's wave function is instantaneously collapsed to the state of ##E_2+c##, and if A is measured to have energy ##E_2##, B's wave function is instantaneously collapsed to the state of ##E_1+c##, so as to keep the total energy of the Universe constant? (##c## is a constant.)When the system is in a superposition of two or more energies, one should think of it as a system that does not have a well defined energy at all. A measurement of energy then gives energy a well defined value. Violation of energy conservation would occur if the system evolved from a state with one value of energy to a state with another value of energy. But here it is not what happens, instead we have an evolution from a state with no value of energy to a state with a value of energy. Hence we cannot say that conservation of energy is violated.
Does this imply that measurements cannot be trusted to give the true energy of a particle? And so the principle of conservation of energy is unfalsifiable?Measurements are not unitary since we’ve ignored the time evolution of the measuring device, hence the apparent “violation”.
No.Does the conservation of energy imply that whenever a particle A is created to be in a superposition of energy ##E_1## and ##E_2##, another particle B must also be created...
No.is it true that, to use @ftr's term, the "excess energy" of A doesn't go to the measuring device in general, but goes to B instead?
No.Does this imply that measurements cannot be trusted to give the true energy of a particle? And so the principle of conservation of energy is unfalsifiable?
How do we know that this is not the case? Particle B could be taken as the rest of the Universe apart from the newly created particle A.No.Does the conservation of energy imply that whenever a particle A is created to be in a superposition of energy ##E_1## and ##E_2##, another particle B must also be created in a superposition of energy ##E_1+c## and ##E_2+c## such that if A is measured to have energy ##E_1##, B's wave function is instantaneously collapsed to the state of ##E_2+c##, and if A is measured to have energy ##E_2##, B's wave function is instantaneously collapsed to the state of ##E_1+c##, so as to keep the total energy of the Universe constant? (##c## is a constant.)
You can't create the rest of the universe in a chosen state. It is what it is. Nor can you measure the energy of the entire universe; you're part of the universe, and such a measurement would require you to be outside it, interacting it with a measuring device also outside it.Particle B could be taken as the rest of the Universe apart from the newly created particle A.
Is QM consistent with the assertion that if a particle A is newly made to be in the superposed state ##\psi_A=c_1\psi_{A, E_1}+c_2\psi_{A, E_2}##, then the rest of the Universe (apart from A) can be prescribed the state ##\psi=c_2\psi_{E_3}+c_1\psi_{E_4}##, where ##E_1+E_3=E_2+E_4##?You can't create the rest of the universe in a chosen state. It is what it is. Nor can you measure the energy of the entire universe; you're part of the universe, and such a measurement would require you to be outside it, interacting it with a measuring device also outside it.
Mathematically, what you wrote down is of course consistent with QM, but there are an infinite number of states we could write down that are mathematically consistent with QM. That doesn't mean they mean anything physically.Is QM consistent with the assertion that if a particle A is newly made to be in the superposed state ##\psi_{A, E_1}+c_2\psi_{A, E_2}##, then the rest of the Universe (apart from A) can be prescribed the state ##\psi=c_2\psi_{E_3}+c_1\psi_{E_4}##, where ##E_1+E_3=E_2+E_4##?
In fact, we don't even know for sure that the concepts "state of the whole universe" and "Hamiltonian operator for the whole universe" are well-defined. All of our experimental knowledge of QM comes from experiments done on limited quantum systems by experimenters and apparatus that are not modeled as part of the quantum system and are considered to be outside it. But, as I said before, there is no way to make measurements or run experiments on the universe from outside it. So we cannot simply take it for granted that we can take the theory developed from measurements "from the outside" and apply it to a system for which there is no "outside".we don't know the state of the whole universe, or even if the whole universe is in some pure state that is an eigenstate of the Hamiltonian operator for the whole universe, even if we don't know exactly which one
Does the principle of conservation of energy (PCoE) rule out the possibility of creating such an entangled pair of particles A and B (where now neither A and B is taken to be the rest of the Universe, to avoid the difficulty in dealing with the rest of the Universe)?Does the conservation of energy imply that whenever a particle A is created to be in a superposition of energy ##E_1## and ##E_2##, another particle B must also be created in a superposition of energy ##E_1+c## and ##E_2+c## such that if A is measured to have energy ##E_1##, B's wave function is instantaneously collapsed to the state of ##E_2+c##, and if A is measured to have energy ##E_2##, B's wave function is instantaneously collapsed to the state of ##E_1+c##, so as to keep the total energy of the Universe constant? (##c## is a constant.)
So is it true that, to use @ftr's term, the "excess energy" of A doesn't go to the measuring device in general, but goes to B instead?
When the system is described as a superposition of two or more eigenstates of a hamiltonian (operator corresponding to the total energy of the studied system).When the system is in a superposition of two or more energies,
What is mathematically E in this mathematical term ##\exp(-\mathrm{i} E t)##? A linear operator?$$|\psi(t) \rangle=\exp(-\mathrm{i} \hat{H} t) |u_E \rangle=\exp(-\mathrm{i} E t) |u_E \rangle.$$
I would have written this instead : ##\langle \hat{H}\rangle_\psi =\langle \psi(t)|\hat{H}|\psi(t) \rangle.##If the system is prepared in a pure state with the state ket ##|\psi(t) \rangle## The expectation value of the energy is given by
$$\langle E \rangle=\langle \psi(t)|\hat{H}|\psi(t) \rangle.$$
No, why would it?Does the principle of conservation of energy (PCoE) rule out the possibility of creating such an entangled pair of particles A and B (where now neither A and B is taken to be the rest of the Universe, to avoid the difficulty in dealing with the rest of the Universe)?
There is no "transfer/teleport of excess energy". Again you are incorrectly limiting your viewpoint to just the two particles, but any measurement on either particle will involve an interaction with a measuring apparatus, and conservation laws will only be satisfied for the full system of particles + apparatus, not for the particles alone, because the particles are not isolated. Conservation of energy (or indeed any conservation law) can only be expected to hold for an isolated system that doesn't interact with anything else.Is it the case that (the existence of) such entangled pair is ruled out because this would cause an instantaneous transfer/teleport of "excess energy" from one part of the Universe (where A is) to another part of the Universe (where B is), violating PCoE locally in both parts of the Universe? Or is it the case that such entangled pair is not rule out because such transfer/teleport of "excess energy" is somehow not actual but only apparent?
That's true if the initial state before those particle were created had a well defined energy. But in general this is not true.Does the conservation of energy imply that whenever a particle A is created to be in a superposition of energy ##E_1## and ##E_2##, another particle B must also be created in a superposition of energy ##E_1+c## and ##E_2+c## such that if A is measured to have energy ##E_1##, B's wave function is instantaneously collapsed to the state of ##E_2+c##, and if A is measured to have energy ##E_2##, B's wave function is instantaneously collapsed to the state of ##E_1+c##, so as to keep the total energy of the Universe constant? (##c## is a constant.)
So is it true that, to use @ftr's term, the "excess energy" of A doesn't go to the measuring device in general, but goes to B instead?