Connection Formulas in the WKB approximation

[maverick280857]

Hello friends,

I've been reading Schiff's book on QM (3rd Edition), esp the section on the WKB approximation. (This isn't homework.)

I have a few questions:

What is the physical significance of the arrow on the connection formulas, like

$$\frac{1}{2}\frac{1}{\sqrt{\kappa}} e^{-\zeta_{2}} \longrightarrow \frac{1}{\sqrt{k}}\cos\left(\zeta_{1}-\frac{\pi}{4}\right)$$

and

$$\frac{e^{-\zeta_{2}}\sin\eta}{\sqrt{\kappa}} \longleftarrow \frac{1}{\sqrt{k}}\cos\left(\zeta_{1}-\frac{\pi}{4}+\eta\right)$$

Notation:

$$k = \sqrt{\frac{2m(E-V(x))}{\hbar^2}}$$
$$\kappa = \sqrt{\frac{2m(V(x)-E)}{\hbar^2}}$$
$$\zeta_{1} = \int_{0}^{x}k(x')dx'$$
$$\zeta_{2} = \int_{x}^{0}\kappa(x')dx'$$

According to the book, these formulas indicate that the wavefunction on the left (in the first formula) goes on to the right with this form, but the converse is not true as the cosine picks up a phase of $\pi/4$. While I've been able to "prove" both these formulas mathematically, I do not have a good feel of what they're saying exactly.

First of all, when I write an arbitrary linear combination of the "exact" solutions to Schrodinger's equation near the turning point (where I have expanded the potential V(x) as a linear function), how do I know what the correct direction of the arrow in the connection formula is?

In other words, after I form the linear combination, I can use the asymptotic forms of the Bessel Function (or the Airy Function) for $x\rightarrow\infty$ and $x\rightarrow -\infty$ thereby obtaining two asymptotic forms of the linear combination. How do I determine whether the arrow points from region 2 (the left of the turning point at x = 0) to region 1 or vice versa?

Thanks for your help.

 

[eljose79]

Dear fellow scientist,

Thank you for sharing your questions and thoughts on the WKB approximation. The connection formulas you mentioned have a physical significance in that they describe the behavior of the wavefunction near the turning point of the potential. The arrow indicates the direction in which the wavefunction is changing as it moves from one region to another. In the first formula, the arrow points from region 2 (left of the turning point) to region 1 (right of the turning point), indicating that the wavefunction decreases as it moves from left to right. In the second formula, the arrow points in the opposite direction, indicating that the wavefunction increases as it moves from right to left.

As for determining the correct direction of the arrow, it depends on the specific potential and the boundary conditions of the problem. In general, the arrow points from the region with higher potential to the region with lower potential. In your case, since the turning point is at x=0, the arrow points from region 2 to region 1. However, if the turning point was at a different location, the direction of the arrow may change accordingly.

I hope this helps clarify the physical significance of the connection formulas. If you have further questions or would like to discuss this topic further, please do not hesitate to reach out.

 

[Entropia]

Hello,

Thank you for your question. The connection formulas in the WKB approximation are used to connect the wavefunction in two different regions, where the potential energy is approximated as a linear function. The arrow in the formulas represents the direction of the wavefunction, i.e. which region it is going from and to. In the first formula, the wavefunction on the left side (region 2) goes to the right side (region 1) with a phase shift of π/4. In the second formula, the wavefunction on the right side (region 1) goes to the left side (region 2) with a phase shift of -π/4.

The physical significance of these formulas lies in the fact that they allow us to approximate the wavefunction in regions where the potential energy is not constant. This is particularly useful in situations where the potential energy changes rapidly, such as near turning points. By connecting the wavefunction in two different regions, we can obtain a more accurate approximation of the wavefunction in the entire region.

To determine the direction of the arrow in the connection formulas, you need to look at the potential energy function. The direction of the arrow will depend on whether the potential energy is increasing or decreasing in that region. If the potential energy is increasing, the arrow points from region 2 to region 1, and if the potential energy is decreasing, the arrow points from region 1 to region 2.

I hope this helps clarify the physical significance of the connection formulas in the WKB approximation. If you have any further questions, please feel free to ask.