Nonrelativistic free particle propagators

In summary, the free particle will move from x to y in time t as U(x,y,t)=\frac{d^3p}{(2\pi)^3}\int\frac{d^3p}{(2\pi)^3}e^{i(p^2/2m)t}e^{i\vec{p}\cdot\Delta x}=\int\frac{d^3p}{(2\pi)^3}e^{i(p^2/2m)t}e^{ip\Delta x\cos\theta}. After performing the angular integrations (which are both trivial), U(x,y,t) will be
  • #1
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This type of integration is a special case of something that occurs over and over in QM and QFT (it's everywhere in Peskin and Schroeder), but I am having a bit of trouble working out the details. Set [itex]\hbar=1[/itex] and consider the propagation amplitude for a free, nonrelativistic particle to move from x to y in time t, given by [itex]U(x,y,t)=\langle y\vert e^{i(p^2/2m)t}\vert x\rangle[/itex]. Peskin and Schroeder evaluate this as a 3-dimensional momentum space integral, and it's easy to manipulate and find [tex]U(x,y,t)=\int\frac{d^3p}{(2\pi)^3}e^{i(p^2/2m)t}e^{i\vec{p}\cdot\Delta x}=\int\frac{d^3p}{(2\pi)^3}e^{i(p^2/2m)t}e^{ip\Delta x\cos\theta}[/tex] where [itex]\Delta \vec x=\vec y -\vec x[/itex]. The next step is to convert to spherical coordinates, which I don't think I have a problem with: [tex]U(x,y,t)=\frac 1{(2\pi)^3}\int_0^\infty dp\, p^2\int_0^{2\pi}d\varphi\int_{-1}^1d\cos\theta e^{ip^2t/2m}e^{ip\Delta x\cos\theta}.[/tex] After performing the angular integrations (which are both trivial) I find [tex]U(x,y,t)=\frac 2{\Delta x(2\pi)^2}\int_0^\infty dp\,p e^{ip^2t/2m}\sin(p\Delta x)[/tex] which not only looks completely intractable but also is nothing like the answer one is supposed to find for a free propagator. Have I screwed this up or is there just some last touch I'm not seeing?
 
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  • #2
To solve this integrale you first note that the integrand is even under ##p\to-p## and therefore you can extend the boundaries up to ##-\infty##, but paying an addition factor of 1/2. Then you recall that ##\sin(p\Delta x)=\text{Im} e^{ip\Delta x}## and once you have done that you're left if an ordinary gaussian integral (complete the square and so on and so forth). Just remember to take the imaginary part at the end of the gaussian integration.
 
  • #3
Thanks! Amazingly the [itex]2\pi[/itex]'s work out using Gaussian integral methods.
 
  • #4
And there should be a minus sign in the exponential with the time!
 

1. What is a nonrelativistic free particle propagator?

A nonrelativistic free particle propagator is a mathematical function that describes the probability amplitude of a particle to travel from one point in space to another in a nonrelativistic system. It takes into account the initial and final positions of the particle, as well as the time it takes for the particle to travel between the two points.

2. How is a nonrelativistic free particle propagator calculated?

A nonrelativistic free particle propagator is calculated using the Feynman path integral, which sums over all possible paths that the particle can take between the initial and final positions. The propagator is then the integral of this sum over all possible times.

3. What is the significance of the nonrelativistic free particle propagator?

The nonrelativistic free particle propagator is an important tool in quantum mechanics as it allows us to calculate the probability of a particle to travel between two points in a nonrelativistic system. It is also used in calculating transition amplitudes and cross sections in scattering experiments.

4. How does the nonrelativistic free particle propagator differ from the relativistic propagator?

The main difference between the nonrelativistic and relativistic free particle propagators lies in the treatment of time. The nonrelativistic propagator assumes a fixed, universal time for all observers, while the relativistic propagator takes into account the time dilation and length contraction effects of special relativity.

5. What are some applications of the nonrelativistic free particle propagator?

The nonrelativistic free particle propagator has various applications in quantum mechanics, such as in calculating the probability of a particle to tunnel through a potential barrier, or in predicting the behavior of particles in a double-slit experiment. It also has practical applications in fields such as material science and quantum computing.

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