Question:
The probability for particle to be at $ \vec{r_2} $ at $ t_2 $ if it was at $ \vec{r_1} $ at $ t_1 $ is given by $ | G(\vec{r_2}, t_2; \vec{r_1}, t_1) |^2 $ . Let us consider a propagator between points $ \vec{r_1} $ and $ \vec{r_2} $ at time t_1 and t_2 which has the form $ G ( \vec{r_2} ,t2; \vec{r_1} , t_1) = 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 $ ( \vec{r_1} , t_1 ) $ to $ (\vec{r_2}, t_2 ) $. What is the probability for the particle to go from $ ( \vec{r_1} , t_1 ) $ to $ (\vec{r_2} , t_2 ) $ ?
Recall that the complex conjugate of a complex number $ z = x + i y $ is $ z^∗ = x − i y $, 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 $ \vec{r_2} $ at $ t_2 $ if it was at $ \vec{r_1} $ at $ t_1 $ is given by $ | G(\vec{r_2}, t_2; \vec{r_1}, t_1) |^2 $ . Let us consider a propagator between points $ \vec{r_1} $ and $ \vec{r_2} $ at time t_1 and t_2 which has the form $ G ( \vec{r_2} ,t2; \vec{r_1} , t_1) = 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 $ ( \vec{r_1} , t_1 ) $ to $ (\vec{r_2}, t_2 ) $. What is the probability for the particle to go from $ ( \vec{r_1} , t_1 ) $ to $ (\vec{r_2} , t_2 ) $ ?
Recall that the complex conjugate of a complex number $ z = x + i y $ is $ z^∗ = x − i y $, 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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