Consider a particle with mass m bound to a potential v(x) = k^2|x|. Using the WKB
quantization condition, find the energy eigenvalues .
To find the energy eigenvalues of a particle bound to a potential v(x) = k^2|x|, we can use the WKB (Wentzel-Kramers-Brillouin) quantization condition. The WKB quantization condition relates the wavefunction of a particle in a potential to the phase integral along a classical turning point.
The WKB approximation assumes that the wavefunction can be written as a product of an amplitude and a phase term, where the phase term varies rapidly compared to the amplitude. The phase integral is then evaluated along the classically allowed region and the turning points, where the potential energy equals the energy of the particle.
Now let's solve for the energy eigenvalues using the WKB quantization condition step by step:
Step 1: Write the Schrödinger equation
Start with the Schrödinger equation for the particle in the potential v(x):
-hbar^2/2m * d^2ψ(x)/dx^2 + v(x) * ψ(x) = E * ψ(x),
where hbar is the reduced Planck's constant, m is the mass of the particle, ψ(x) is the wavefunction, E is the energy, and v(x) is the potential energy.
Step 2: Separate the Schrödinger equation
Rearrange the equation to isolate the second derivative term:
d^2ψ(x)/dx^2 = -((2m / hbar^2) * (E - v(x))) * ψ(x).
Step 3: Apply the WKB approximation
In the WKB approximation, we can write ψ(x) as ψ(x) = A(x) * e^(iS(x)/hbar), where A(x) is the amplitude and S(x) is the phase.
Step 4: Substitute the WKB wavefunction into the Schrödinger equation
Substitute ψ(x) = A(x) * e^(iS(x)/hbar) into the Schrödinger equation and separate the amplitude and phase terms.
Step 5: Calculate the phase integral
Integrate the phase term S(x) along the classically allowed region, which is between the turning points where E = v(x). This gives us the phase integral I.
Step 6: Apply the WKB quantization condition
The WKB quantization condition states that the phase integral I is quantized. This can be written as:
I = (n + 1/2) * h, where n is an integer and h is Planck's constant.
Step 7: Simplify the phase integral
Simplify the phase integral using the expression for S(x) and differentiate it with respect to x. Then, evaluate the resulting expression at the turning points.
Step 8: Solve for the energy eigenvalues
By equating the two expressions for the phase integral, we can solve for the energy eigenvalues E.
It's important to note that the WKB approximation is an approximate method and might not yield exact results for all cases. However, it can provide good approximations for certain potentials, including the one given in this question.
To obtain the exact energy eigenvalues for the potential v(x) = k^2|x|, it would be more accurate to solve the Schrödinger equation using other methods, such as numerical techniques or approximation methods specific to piecewise-defined potentials.