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Path integral with initial definite momentum

Physics Asked on June 10, 2021

Generally, in path integral formalism a propagator between two states with definite position is computed, something like,
$$K(x_1,t_1;x_0,t_0)=int_{x_0(t_0)}^{x_1(t_1)}mathcal{D}x(t)expleft(frac{i}{hbar}S[x(t)]right).$$
However, if we want a propagator between two states, where the initial state has a definite momentum and final state has definite position, that is,
$$K(x_1,t_1;p_0,t_0),$$
then how does one goes on compute such a quantity? Are there books/notes discussing this? Can someone construct this for the free non-relativistic particle case?

One Answer

$$ K(x_1,t_1;p_0,t_0)~=~frac{1}{sqrt{2pihbar}} int_{mathbb{R}} ! dx_0~ K(x_1,t_1;x_0,t_0)expleft(frac{ip_0x_0}{hbar}right), $$ cf. e.g. this Phys.SE post.

Answered by Qmechanic on June 10, 2021

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