- #1
AntiStrange
- 20
- 1
I'm reading up on Quantum Mechanics and I don't follow an integration they use.
They start with this:
[tex]\psi(x,t) = \int^{\infty}_{-\infty} dk A(k) e^{i(kx-\omega t)}[/tex]
They begin by considering the wave packet at time t=0:
[tex]\psi(x,0) = \int^{\infty}_{-\infty} dk A(k) e^{ikx}[/tex]
"and illustrate it by considering a special form, called the gaussian form":
[tex]A(k) = e^{-\alpha (k-k_{0})^{2} / 2}[/tex]
I'm ok with all of this so far, although not entirely sure why they chose (or what even the purpose is of) the "gaussian form". I do know a very little bit about the gaussian distribution and I see that this "gaussian form" looks a little similar to a portion of the noramal distribution but why just that part, and why the differences. But, at any rate I can live with all of that so far, however I get completely lost once they do the next step.
They make a change of variables to q' = k-k₀, and end up with:
[tex]\psi(x,0) = e^{ik_{0}x}e^{-x^{2} / 2\alpha} \int^{\infty}_{-\infty} dq' e^{-\alpha q'^{2} / 2}[/tex]
[tex]= \sqrt{\frac{2\pi}{\alpha}}e^{ik_{0}x}e^{-x^{2} / 2\alpha}[/tex]
Any help, advice, insights, or even just a point in the right direction would really help me right now. Thanks.
They start with this:
[tex]\psi(x,t) = \int^{\infty}_{-\infty} dk A(k) e^{i(kx-\omega t)}[/tex]
They begin by considering the wave packet at time t=0:
[tex]\psi(x,0) = \int^{\infty}_{-\infty} dk A(k) e^{ikx}[/tex]
"and illustrate it by considering a special form, called the gaussian form":
[tex]A(k) = e^{-\alpha (k-k_{0})^{2} / 2}[/tex]
I'm ok with all of this so far, although not entirely sure why they chose (or what even the purpose is of) the "gaussian form". I do know a very little bit about the gaussian distribution and I see that this "gaussian form" looks a little similar to a portion of the noramal distribution but why just that part, and why the differences. But, at any rate I can live with all of that so far, however I get completely lost once they do the next step.
They make a change of variables to q' = k-k₀, and end up with:
[tex]\psi(x,0) = e^{ik_{0}x}e^{-x^{2} / 2\alpha} \int^{\infty}_{-\infty} dq' e^{-\alpha q'^{2} / 2}[/tex]
[tex]= \sqrt{\frac{2\pi}{\alpha}}e^{ik_{0}x}e^{-x^{2} / 2\alpha}[/tex]
Any help, advice, insights, or even just a point in the right direction would really help me right now. Thanks.