Suppose a particle with a definite energy is going in empty space from a location r 1 to a location r 2 . In other words, it is a free particle with no forces on it. Except for a numerical factor in front, the amplitude to go from r 1 to r 2 is (3.7) ⟨ r 2 | r 1 ⟩ = e i ℏ p r 12 r 12 .

Question

Suppose a particle with a definite energy is going in empty space from a location       r    1   to a location       r    2  . In other words, it is a free particle with no forces on it. Except for a numerical factor in front, the amplitude to go from       r    1   to       r    2   is                    (3.7)                    ⟨                  r          2                          |                          r          1                ⟩        =                              e                                          i                ℏ                            p                              r                                  12                                                                          r                          12                                      .

Rachael Trevino · Accepted Answer

Feynman&#039;s expression                    (3.7)                    G        (                  r                )                 =                 ⟨                              r                    2                          |                                      r                    1                ⟩                 =                                       exp            ⁡            (            i            k            r            )                    r                    is the propagator/Greens function  (      ∇    2    +      k    2    )  G  (      r    )     =     −  4  π      δ    3    (      r    )for the 3D Helmholtz equation/the free TISE.

Suppose a particle with a definite energy is going in empty space from a location r_1 to a location r_2. In other words, it is a free particle with no forces on it. Except for a numerical factor in front, the amplitude to go from r_1 to r_2

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