- #1

xebtl2

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Hi,

Note: I will be sloppy with constant factors in this post. Only the general structure of the equations matters.

Consider a particle in a linear potential,

Mathematically, this is a second-order ODE, and there are two solutions, related to the Airy functions Ai and Bi. Physically, only the Ai solution has the correct boundary condition; Ai(y) drops exponentially to zero for y > 0, whereas Bi(y) diverges exponentially. So one is led to select the Ai solution as the physical one and discard the Bi solution.

Now, let's go to momentum space, where

By the familiar properties of the Fourier transform, d

The question that occurs to me is, what happened to the second solution? Why did it disappear when going to momentum space, and how was the Fourier transform able to pick out the physical solution? Where is the Airy of yesteryear?

Additional notes:

Note: I will be sloppy with constant factors in this post. Only the general structure of the equations matters.

Consider a particle in a linear potential,

[itex]\frac{\mathrm d^2}{\mathrm d x^2} \psi(x) + x \psi(x) - E \psi(x) = 0.[/itex]

Mathematically, this is a second-order ODE, and there are two solutions, related to the Airy functions Ai and Bi. Physically, only the Ai solution has the correct boundary condition; Ai(y) drops exponentially to zero for y > 0, whereas Bi(y) diverges exponentially. So one is led to select the Ai solution as the physical one and discard the Bi solution.

Now, let's go to momentum space, where

[itex]\frac{\mathrm d}{\mathrm d k} \psi(k) + k^2 \psi(k) - E \psi(k) = 0.[/itex]

By the familiar properties of the Fourier transform, d

^{2}/dx^{2}became a k^{2}, and x became a d/dk. We are left with a*first-order*ODE, which has*one*linearly independent solution, namely the physical one corresponding to Ai.The question that occurs to me is, what happened to the second solution? Why did it disappear when going to momentum space, and how was the Fourier transform able to pick out the physical solution? Where is the Airy of yesteryear?

Additional notes:

- Both Ai and Bi seem to have well-defined Fourier transforms, see http://functions.wolfram.com/Bessel-TypeFunctions/AiryAi/22/01/ and http://functions.wolfram.com/Bessel-TypeFunctions/AiryBi/22/01/.
- This stated problem is the one that brought up the question, but I could ask a very similar one about Dirac's treatment of the harmonic oscillator.

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