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Hi all, I am rather confused about the following concept. Assistance is greatly appreciated!

A time-dependent probability amplitude can be written as

$$\langle a_k| e^{-\frac{i}{\hbar}\hat{H}t} |\psi\rangle$$

where ##a_k## is an eigenvalue. Suppose I want the x-representation of the ket, I can then choose to write this as

$$\langle x,t |\psi\rangle$$

or as

$$\langle x |\psi(t)\rangle$$

While I understand that either way is equally valid, and makes no difference to the physics, how does it change the way I interpret things?

For example, if I was observing how some atom prepared in state ##|\psi\rangle## propagates in space, could I interpret the ##\langle x |\psi(t)\rangle## case as me having my apparatus fixed at one point in space, while the particle evolves (propagates relative to my apparatus) as time evolves?

What then, would be the interpretation for the ##\langle x,t |\psi\rangle## case? Is this related to the idea that the operator/observable evolves, rather than the state (though I don't really understand what this means either).

A time-dependent probability amplitude can be written as

$$\langle a_k| e^{-\frac{i}{\hbar}\hat{H}t} |\psi\rangle$$

where ##a_k## is an eigenvalue. Suppose I want the x-representation of the ket, I can then choose to write this as

$$\langle x,t |\psi\rangle$$

or as

$$\langle x |\psi(t)\rangle$$

While I understand that either way is equally valid, and makes no difference to the physics, how does it change the way I interpret things?

For example, if I was observing how some atom prepared in state ##|\psi\rangle## propagates in space, could I interpret the ##\langle x |\psi(t)\rangle## case as me having my apparatus fixed at one point in space, while the particle evolves (propagates relative to my apparatus) as time evolves?

What then, would be the interpretation for the ##\langle x,t |\psi\rangle## case? Is this related to the idea that the operator/observable evolves, rather than the state (though I don't really understand what this means either).

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