WKB approximation

The Wentzel-Kramers-Brillouin (WKB) approximation is a powerful method used in quantum mechanics to solve the Schrödinger equation for a single particle. This approximation is particularly useful for understanding systems where the potential changes slowly, such as in quantum tunneling and in the semi-classical analysis of quantum systems.

Basic concept

Let's start with the time-independent Schrödinger equation in one dimension:

        -ħ²/2m * (d²ψ/dx²) + V(x)ψ(x) = Eψ(x)
    

Here, ħ is the reduced Planck constant, m is the mass of the particle, ψ(x) is the wave function, V(x) is the potential energy, and E is the total energy of the particle.

The WKB approximation applies when the potential V(x) changes slowly in space. The basic idea is to express the wave function ψ(x) as an exponential function whose exponent is an integral of a parameter related to the classical motion of the particle.

Etymology

The form of the wave function is assumed to be as follows:

        ψ(x) = A(x) exp(i S(x) / ħ)
    

where A(x) is the amplitude and S(x) is the action. Substituting this form into the Schrödinger equation and applying the WKB approximation, we find the following expression for S(x) :

        dS(x)/dx = ±√(2m(E - V(x)))
    

This equation tells us that S(x) is related to the classical motion. Integrating gives:

        S(x) = ∫ ±√(2m(E - V(x))) dx
    

Regions and milan

To use the WKB approximation, it is important to consider the behavior of the wave function in different regions of the potential:

• Classically accepted region (E > V(x)): In this region, the particle can exist. The wave function has the form:

                  ψ(x) = A(x) exp(±i ∫ √(2m(E - V(x))) dx / ħ)
              

• Classically forbidden region (E < V(x)): the presence of the particle is only supported by quantum mechanics (like tunneling). The wave function is:

                  ψ(x) = B(x) exp(± ∫ √(2m(V(x) - E)) dx / ħ)
              

Turning point

An inflection point occurs when E = V(x). At inflection points, the WKB approximation breaks down because the parameter inside the integral goes to zero. We need to carefully match the solutions on both sides. Consider possibilities such as a simple harmonic oscillator or a quantum well, where the concept of inflection points is relevant.

In the above diagram, you see a potential curve and a horizontal line representing the energy E The points where the curve crosses this line are the inflection points.

Connection formula

Connection formulas express the behavior of the wave function near the turning point. They ensure that solutions in the classically allowed and forbidden regions are smoothly connected. This is important when calculating phenomena such as the tunneling probability. Normally, we use Airy functions to deal with solutions near the turning point.

Example: Harmonic oscillator

For example, consider the quantum harmonic oscillator. Its potential is:

        V(x) = 1/2 m ω² x²
    

Conventionally, turning points are located where the potential energy is equal to the total energy:

        E = 1/2 m ω² x²
    

Using WKB, we estimate different fields and match them according to the rules of quantum mechanics.

Example: Rectangular potential barrier

Another important application of the WKB approximation is to understand quantum tunneling, as seen in a rectangular potential barrier:

        V(x) = { 0 if x < 0, V₀ if 0 ≤ x ≤ a, 0 if x > a }
    

For a particle moving towards a barrier with energy E such that E < V₀, classical physics predicts no path, but quantum mechanics allows tunneling. It is important here to understand the exponential decay of the wave function inside the barrier.

Applications and limitations

The WKB approximation is widely used in quantum mechanics as it provides a semi-classical approach and simplifies complex problems. It is also used in the study of gravitational wave phenomena in atomic physics, quantum chemistry, and even astrophysics.

However, the approximation has its limitations. It fails near turning points and cannot take into account abrupt changes in the potential. In cases of strong quantum effects, such as those involving very low energies or potentials with sharp features, more accurate methods are preferable.

Conclusion

The WKB approximation is a versatile tool in an advanced quantum mechanics toolkit, bridging the gap between classical mechanics and the wave-like nature of quantum mechanics. While it simplifies the understanding of quantum systems in slowly changing probabilities, caution must be exercised around inflection points and the limitations inherent in any approximation must be considered.

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