How does one understand the law of energy conservation in quantum mechanics?
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In physics, the law of conservation of energystates that the total energy of an isolated system remains constant, it is said to be conserved over time.[1] This law means that energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another. For instance, chemical energy is converted to kinetic energy when a stick of dynamiteexplodes. If one adds up all the forms of energy that were released in the explosion, such as the kinetic energy of the pieces, as well as heat and sound, one will get the exact decrease of chemical energy in the combustion of the dynamite. Classically, conservation of energy was distinct from conservation of mass; however, special relativity showed that mass is related to energy and vice versa by E = mc2, and science now takes the view that mass–energy is conserved.
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There are several problems with the phrase in italics:
First of all, this is not "Heisenberg uncertainty". The Heisenberg uncertainty principle, as it is understood nowadays, involves the variances of two observables, which are not jointly measureable. Since time is no observable in quantum mechanics (neither in quantum field theory), this equation is not the "Heisenberg uncertainty principle". It is only often referred to as such, because it looks like the original formula. So much for the semantics.
Second, there is no problem with the conservation of energy. There also shouldn't be, because we have time translation invariance and hence should obtain conservation of energy by Noether (roughly speaking). So, energy is conserved within the framework of quantum mechanics. Can we see this? Yes, the unitary evolution of the state commutes with the Hamiltonian (as it is defined as exponential of the Hamiltonian) and thus the energy, which is the expectation value of the Hamiltonian, stays constant over all times.
Note also that you are talking about expectation values (since that's all we can do). The above energy-time uncertainty tells us something about the limits of measurements and preparations. Having a state with an energy E, if we measure this state, we will only be able to determine its energy up to some precision - which is limited by the amount of time we observe the particle. Losly speaking: If I only take a quick look, my measurement will likely be off. Similarly, a state living only a short time, will not have a well-defined energy.
Third, you mention a process in particle physics. It's true, your energy-time uncertainty gets mentioned a lot in quantum field theory and people like to interpret it as short time violation of energy conservation, but to my understanding, that's just not true. The problem is, that all these calculations (and the corresponding diagrams) come out of perturbation theory and if you have a look at nonperturbative exact calculations, the effects are gone - hence they are artifacts of perturbation theory. We just like to interpret them like this, because it gives a meaning to our calculations. In this vein, since all our "off-shell" particles are called "virtual particles", one should call the "borrowing" of energy a "virtual violation".
hope it helps you
please mark as brainiest answer
¶potter¶
First of all, this is not "Heisenberg uncertainty". The Heisenberg uncertainty principle, as it is understood nowadays, involves the variances of two observables, which are not jointly measureable. Since time is no observable in quantum mechanics (neither in quantum field theory), this equation is not the "Heisenberg uncertainty principle". It is only often referred to as such, because it looks like the original formula. So much for the semantics.
Second, there is no problem with the conservation of energy. There also shouldn't be, because we have time translation invariance and hence should obtain conservation of energy by Noether (roughly speaking). So, energy is conserved within the framework of quantum mechanics. Can we see this? Yes, the unitary evolution of the state commutes with the Hamiltonian (as it is defined as exponential of the Hamiltonian) and thus the energy, which is the expectation value of the Hamiltonian, stays constant over all times.
Note also that you are talking about expectation values (since that's all we can do). The above energy-time uncertainty tells us something about the limits of measurements and preparations. Having a state with an energy E, if we measure this state, we will only be able to determine its energy up to some precision - which is limited by the amount of time we observe the particle. Losly speaking: If I only take a quick look, my measurement will likely be off. Similarly, a state living only a short time, will not have a well-defined energy.
Third, you mention a process in particle physics. It's true, your energy-time uncertainty gets mentioned a lot in quantum field theory and people like to interpret it as short time violation of energy conservation, but to my understanding, that's just not true. The problem is, that all these calculations (and the corresponding diagrams) come out of perturbation theory and if you have a look at nonperturbative exact calculations, the effects are gone - hence they are artifacts of perturbation theory. We just like to interpret them like this, because it gives a meaning to our calculations. In this vein, since all our "off-shell" particles are called "virtual particles", one should call the "borrowing" of energy a "virtual violation".
hope it helps you
please mark as brainiest answer
¶potter¶
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