WKB Quantization for Bound States

Hello! Welcome to our final lesson on the WKB approximation.

In our previous lessons, we've established the WKB method as a powerful tool for finding approximate solutions to the Schrödinger equation. We derived the WKB wavefunctions, understood the need for connection formulae at classical turning points, and applied this machinery to calculate tunneling probabilities for unbound particles.

Today, we shift our focus from unbound states to bound states. Our goal is to use the WKB approximation to find the allowed energy levels for a particle trapped in a potential well. By the end of this lesson, you will be able to derive the Bohr-Sommerfeld quantization condition for bound states using the WKB method. This famous result was a cornerstone of the "old quantum theory" and the WKB method provides its modern justification.

1. The Physical Picture: Wave Self-Consistency

Before diving into the mathematics, let's build some physical intuition. Consider a particle with energy $E$ trapped in a one-dimensional potential well, oscillating between two classical turning points, $x_1$ and $x_2$.

In the classically allowed region, the WKB solution is a superposition of a right-moving and a left-moving wave. For a stationary bound state to exist, the wavefunction must be self-consistent. Imagine a wave starting at some point, moving to the right. It reflects from the turning point $x_2$, travels back to the left, reflects from $x_1$, and returns to its starting point. For a stable standing wave to form, the total phase accumulated during this round trip must be an integer multiple of $2\pi$.

The total phase has two components:

1. Dynamical Phase: This is the phase accumulated from the particle's momentum, given by $\frac{1}{\hslash }\oint p(x)dx$, where the integral is over one full classical orbit.
2. Phase Shift on Reflection: As the wave reflects from a turning point, it experiences a phase shift. In the previous lesson, we saw that connecting an oscillating solution to a decaying exponential across a turning point involves a phase factor of $\pi /4$. Since there are two turning points in a potential well, we can anticipate a total phase shift of $2\times (-\pi /2)=-\pi$ for a full round trip.

Combining these, the condition for a stationary state becomes:
$\frac{1}{\hslash }\oint p(x)dx-\pi =2n\pi ,n=0,1,2,\dots$
Rearranging this gives the celebrated Bohr-Sommerfeld quantization condition:
$\oint p(x)dx=(n+\frac{1}{2})2\pi \hslash =(n+\frac{1}{2})h$
This intuitive argument is powerful, and now we will prove it rigorously using the connection formulae.

2. Deriving the Quantization Condition

The formal derivation hinges on ensuring the wavefunction is well-behaved everywhere. Specifically, it must decay to zero deep inside the classically forbidden regions on either side of the well. We will enforce this boundary condition and demand that the wavefunction in the central, classically allowed region is single-valued.

The logic is as follows:

1. Start in the left forbidden region ($x<x_1$), where the wavefunction must be a decaying exponential.
2. Use the appropriate connection formula to find the form of the wavefunction in the allowed region ($x_1<x<x_2$).
3. Independently, start in the right forbidden region ($x>x_2$), where the wavefunction must also be a decaying exponential.
4. Use the connection formula at $x_2$ to find a second form for the wavefunction in the allowed region.
5. These two expressions for the wavefunction in the central region must be identical. This consistency requirement will constrain the allowed energies, leading to the quantization condition.

This derivation is a classic and important piece of physics. The following video provides a clear, step-by-step walkthrough of the mathematical details.

Let's summarize the key steps from the video.

• In the allowed region II ($x_1<x<x_2$), connecting the solution to the decaying exponential in region I ($x<x_1$) gives a wavefunction of the form:
  ${\psi }_{II}(x)\propto \frac{1}{\sqrt{p(x)}}\cos \left(\frac{1}{\hslash }{\int }_{x_1}^{x}p(x^{\prime })d{x}^{\prime }-\frac{\pi }{4}\right)$
• Similarly, connecting to the decaying exponential in region III ($x>x_2$) gives a wavefunction of the form:
  ${\psi }_{II}(x)\propto \frac{1}{\sqrt{p(x)}}\cos \left(\frac{1}{\hslash }{\int }_x^{x_2}p(x^{\prime })d{x}^{\prime }-\frac{\pi }{4}\right)$
• For these two expressions to represent the same physical state, their arguments must be equal, up to a sign and an integer multiple of $\pi$. Choosing the same sign, we have:
  $\frac{1}{\hslash }{\int }_{x_1}^{x}p(x^{\prime })d{x}^{\prime }-\frac{\pi }{4}=\pm \left(\frac{1}{\hslash }{\int }_x^{x_2}p(x^{\prime })d{x}^{\prime }-\frac{\pi }{4}-n\pi \right)$
  The choice of sign and integer $n$ can be a bit subtle, but requiring the functions to match for all $x$ leads to the condition:
  $\frac{1}{\hslash }\left({\int }_{x_1}^{x}p(x^{\prime })d{x}^{\prime }+{\int }_x^{x_2}p(x^{\prime })d{x}^{\prime }\right)-\frac{\pi }{2}=n\pi$
• Combining the integrals gives the final result:
  ${\int }_{x_1}^{x_2}p(x^{\prime })d{x}^{\prime }=\left(n+\frac{1}{2}\right)\pi \hslash$
  Since $\oint p(x)dx=2{\int }_{x_1}^{x_2}p(x^{\prime })dx$, this is precisely the Bohr-Sommerfeld condition we anticipated. This equation implicitly defines the quantized energy levels $E_n$, because the momentum $p(x)=\sqrt{2m(E-V(x))}$ and the turning points $x_1,x_2$ all depend on $E$.

3. Application: The Harmonic Oscillator

Let's test this formula on a system where we know the exact answer: the simple harmonic oscillator, $V(x)=\frac{1}{2}m{\omega }^2{x}^2$. This is a crucial benchmark for the WKB method.

As the notes demonstrate, the calculation proceeds as follows:

1. Turning Points: For an energy $E$, the turning points are where $E=\frac{1}{2}m{\omega }^2{x}^2$, which gives $x_{1,2}=\mp \sqrt{2E/m{\omega }^2}$.
2. The Integral: We must solve ${\int }_{x_1}^{x_2}\sqrt{2m(E-\frac{1}{2}m{\omega }^2{x}^2)}dx$.
   This integral is standard and evaluates to $\frac{\pi E}{\omega }$.
3. Quantization: Applying the condition:
   $\frac{\pi E_n}{\omega }=\left(n+\frac{1}{2}\right)\pi \hslash$
4. Energy Levels: Solving for $E_n$ gives:
   $E_n=\left(n+\frac{1}{2}\right)\hslash \omega$

This is the exact energy spectrum of the quantum harmonic oscillator! This is a remarkable result. For most potentials, the WKB approximation is only accurate for large quantum numbers $n$, but for the harmonic oscillator, it happens to be exact for all $n$. This "fluke" provides strong validation for the method.

4. Validity and Interpretation

The half-integer quantization, $(n+1/2)$, is a significant improvement over the integer quantization, $n$, of the old Bohr-Sommerfeld theory. The extra $\frac{1}{2}$ comes directly from the $\pi /4$ phase shifts at each of the two "soft" turning points where the potential is smooth. This term is responsible for the correct zero-point energy of the harmonic oscillator ($E_0=\frac{1}{2}\hslash \omega$).

What if the turning points are not "soft"? Consider a particle in an infinite square well from $x=0$ to $x=L$. The potential walls are infinitely steep ("hard" turning points). A wave reflecting from a hard wall undergoes a phase shift of $-\pi$. The round trip phase shift is therefore $-2\pi$.
The quantization condition becomes:
$\frac{1}{\hslash }\oint pdx-2\pi =2n\pi ⟹2pL=(n+1)2\pi \hslash$
With $N=n+1=1,2,...$, we get $p=N\pi \hslash /L$, leading to the exact energy levels $E_N=\frac{p^2}{2m}=\frac{N^2{\pi }^2{\hslash }^2}{2m{L}^2}$. The key is that the phase shifts at the turning points are crucial and depend on the nature of the potential.

Conclusion

This lesson concludes our study of the WKB approximation. We have seen how it can be applied to both unbound (tunneling) and bound states.

Key Takeaways:

• The requirement that a bound state wavefunction be self-consistent in a potential well leads to a quantization of energy levels.
• The WKB method provides a formal derivation for this, resulting in the Bohr-Sommerfeld quantization condition: $\oint p(x)dx=(n+1/2)h$.
• This condition states that the area of the classical orbit in phase space is quantized in half-integer units of Planck's constant, $h$.
• The $\frac{1}{2}$ factor arises from phase shifts at the classical turning points and is crucial for obtaining correct results, like the zero-point energy.
• The WKB approximation is generally most accurate for large quantum numbers but can sometimes yield exact results for special potentials like the harmonic oscillator.

Preview of the Next Lesson:
We have now completed our survey of key approximation methods for stationary and time-dependent problems. We will now move to a new major topic: Quantum Scattering Theory. We will begin by developing the formal framework for describing how quantum particles scatter off a potential, starting with the derivation of the fundamental Lippmann-Schwinger equation. This will lay the groundwork for understanding everything from particle physics experiments to electron transport in your field of condensed matter.