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pellman
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Typical introductions to path integrals start with asking for the value of [itex]\langle x_1,t_1 | x_2,t_2 \rangle[/itex]. This is usually interpreted as the probability amplitude of observing a particle at x_2 at time time t_2 given that it is located at x_1 at t_1.
But is this so? I am having trouble relating this to the theory in terms of physically realizable states. Exact position states are not normalizable. So suppose we have a wave function [itex]\Psi (x,t)[/itex] which is a solution to our Schrodinger equation and satisfies the desired initial conditions. It can be considered as the expansion of an abstract state [itex]|\Psi(t)\rangle[/itex] in the basis [itex]|x\rangle[/itex], which are eigenstates of the (time-independent) position operator, so that [itex]\Psi (x,t)=\langle x|\Psi(t)\rangle[/itex]. The time-dependence can be shifted to the basis states by going to the Heisenberg picture so that we have basis states [itex]|x,t\rangle[/itex] that are eigenstates of the time-dependent Heisenberg picture position operator. [itex]\Psi (x,t)=\langle x,t|\Psi\rangle[/itex]. Both cases can be expressed as [itex]\langle x|e^{-iHt}|\Psi\rangle[/itex].
But when we ask the question of quantum observation, we don't get something which looks like [itex]\langle x_1,t_1 | x_2,t_2 \rangle[/itex]. Which is: given a system in a state corresponding to wave function [itex]\Psi (x,t_1)[/itex] at time t_1, what is the probability amplitude of observing the system in state [itex]\Phi(x,t_2)[/itex] at time t_2?
The answer is
[tex]\int \Phi(x,t_2)^* \Psi(x,t_2)d^3 x[/tex]
But t_1 doesn't enter into this calculation, at least not directly. It only enters as the initial condition for our solution Psi of the Schrodinger equation. Once we have that solution, we know the state time-evolved to t_2 and then calculate the inner product at that time t_2. (and if we know the time-dependence of [itex]\Phi[/itex] we can calculate this integral at any time. It is time-independent.)
So what sort of question about physically realizable states is [itex]\langle x_1,t_1 | x_2,t_2 \rangle[/itex] the answer to? or what question does [itex]\langle x_1,t_1 | x_2,t_2 \rangle[/itex] help us solve?
Or, even, what is a quantum question which depends on a time difference t_2 - t_1 ?
But is this so? I am having trouble relating this to the theory in terms of physically realizable states. Exact position states are not normalizable. So suppose we have a wave function [itex]\Psi (x,t)[/itex] which is a solution to our Schrodinger equation and satisfies the desired initial conditions. It can be considered as the expansion of an abstract state [itex]|\Psi(t)\rangle[/itex] in the basis [itex]|x\rangle[/itex], which are eigenstates of the (time-independent) position operator, so that [itex]\Psi (x,t)=\langle x|\Psi(t)\rangle[/itex]. The time-dependence can be shifted to the basis states by going to the Heisenberg picture so that we have basis states [itex]|x,t\rangle[/itex] that are eigenstates of the time-dependent Heisenberg picture position operator. [itex]\Psi (x,t)=\langle x,t|\Psi\rangle[/itex]. Both cases can be expressed as [itex]\langle x|e^{-iHt}|\Psi\rangle[/itex].
But when we ask the question of quantum observation, we don't get something which looks like [itex]\langle x_1,t_1 | x_2,t_2 \rangle[/itex]. Which is: given a system in a state corresponding to wave function [itex]\Psi (x,t_1)[/itex] at time t_1, what is the probability amplitude of observing the system in state [itex]\Phi(x,t_2)[/itex] at time t_2?
The answer is
[tex]\int \Phi(x,t_2)^* \Psi(x,t_2)d^3 x[/tex]
But t_1 doesn't enter into this calculation, at least not directly. It only enters as the initial condition for our solution Psi of the Schrodinger equation. Once we have that solution, we know the state time-evolved to t_2 and then calculate the inner product at that time t_2. (and if we know the time-dependence of [itex]\Phi[/itex] we can calculate this integral at any time. It is time-independent.)
So what sort of question about physically realizable states is [itex]\langle x_1,t_1 | x_2,t_2 \rangle[/itex] the answer to? or what question does [itex]\langle x_1,t_1 | x_2,t_2 \rangle[/itex] help us solve?
Or, even, what is a quantum question which depends on a time difference t_2 - t_1 ?
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