Statement
Suppose we have a (2+1)-dimensional spacetime. I have just detected a quantum particle at the position $A=(x, y) = (0, 0)$ with time $t = 0$. Obviously, at time $t = 0$, the probability of finding it at a different position, such as $(2, 6)$, is 0. But what is the probability of detecting the particle at $B=(2, 6)$ at $t = 2$? Assume $m = 1$.
Development
To calculate the probability of detecting a quantum particle at a different position at a later time, you would generally use the principles of quantum mechanics, specifically the wave function evolution. Since you're dealing with a quantum particle, its behavior is governed by the Schrodinger equation:
$$ i\hbar \frac{\partial}{\partial t} \Psi(x, y, t) = \hat{H} \Psi(x, y, t). $$To solve this problem, you would:
1. Define the Initial State: Determine the wave function $\Psi(x, y, 0)$ at $t=0$. Since you detected the particle at $A=(0,0)$ at $t=0$, the wave function would be sharply peaked around this point. A Dirac delta, or something like that.
2. Set Up the Hamiltonian: For a particle with mass $m=1$, in a potential $V(x, y)$, the Hamiltonian is typically $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(x, y)$. You would need to know the potential energy function $V(x, y)$ to proceed.
3. Evolve the Wave Function: Use the Schrödinger equation to evolve the wave function from $t=0$ to $t=2$. This involves solving partial differential equations, which can be complex and may require numerical methods.
4. Calculate the Probability: The probability of finding the particle at position $B=(2,6)$ at $t=2$ is given by the square of the magnitude of the wave function at that point and time: $P(2,6,2) = |\Psi(2, 6, 2)|^2$.
Aside question
What if I want to compute the probability of being at $B=(2,6)$ but going first through $C=(1,1)$ at time $t=1$? How should I proceed?
Answer
1. Compute the Amplitude for $(1,1)$ at $t=1$: First, you would calculate the probability amplitude for the particle to move from $(0,0)$ at $t=0$ to $(1,1)$ at $t=1$. This involves solving the time-dependent Schrödinger equation as before.
2. Compute the Amplitude for $(2,6)$ at $t=2$: Next, calculate the probability amplitude for moving from $(1,1)$ at $t=1$ to $(2,6)$ at $t=2$. Again, this involves the Schrödinger equation for this specific leg of the journey.
3. Multiply the Amplitudes: The total amplitude for the path going through $(1,1)$ at $t=1$ is the product of the amplitudes of the two legs of the journey. Why? I don't understand this yet. See section 2 in Feynman "Space-time approach to non-relativistic quantum mechanics".
4. Calculate the Probability: The probability of the particle being at $(2,6)$ at $t=2$, given it passed through $(1,1)$ at $t=1$, is the square of the magnitude of the total amplitude calculated in the previous step.
I think these two questions are fundamental to understand Feynman's path integral formulation.
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Author of the notes: Antonio J. Pan-Collantes
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