Question:
The probability for particle to be at →r2 at t2 if it was at →r1 at t1 is given by |G(→r2,t2;→r1,t1)|2 . Let us consider a propagator between points →r1 and →r2 at time t_1 and t_2 which has the form G(→r2,t2;→r1,t1)=a+b, where a and b are complex and non-zero. This can, in some sense, be considered the case where there are two (and only two) possible ways for particle to travel from (→r1,t1) to (→r2,t2). What is the probability for the particle to go from (→r1,t1) to (→r2,t2) ?
Recall that the complex conjugate of a complex number z=x+iy is z∗=x−iy, where x and y are real numbers.
Solution:
This is a really complicated way to simply asking what is |a+b|2, We can simply compute that as |a+b|2=(a+b)(a+b)∗=(a+b)(a∗+b∗)=aa∗+ab∗+ba∗+bb∗=|a|2+ab∗+ba∗+|b|2.
The probability for particle to be at →r2 at t2 if it was at →r1 at t1 is given by |G(→r2,t2;→r1,t1)|2 . Let us consider a propagator between points →r1 and →r2 at time t_1 and t_2 which has the form G(→r2,t2;→r1,t1)=a+b, where a and b are complex and non-zero. This can, in some sense, be considered the case where there are two (and only two) possible ways for particle to travel from (→r1,t1) to (→r2,t2). What is the probability for the particle to go from (→r1,t1) to (→r2,t2) ?
Recall that the complex conjugate of a complex number z=x+iy is z∗=x−iy, where x and y are real numbers.
Solution:
This is a really complicated way to simply asking what is |a+b|2, We can simply compute that as |a+b|2=(a+b)(a+b)∗=(a+b)(a∗+b∗)=aa∗+ab∗+ba∗+bb∗=|a|2+ab∗+ba∗+|b|2.
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