The Quantum Particle in a Linear Potential is a rigorous test bed for understanding quantum confinement and approximation methods. Physically, it represents a particle under a constant force, such as a "quantum bouncer" under gravity. Mathematically, it bridges the gap between elementary differential equations and special functions (Airy functions). This post provides the complete, step-by-step derivation of the exact solution and the WKB approximation without skipping any algebraic details.
Figure 1: The symmetric linear potential \( V(x) = k|x| \) showing the first few discrete energy levels and their wavefunctions.
Full Derivation: Quantum Particle in a Linear Potential
1. System Definition and Hamiltonian
We consider a particle of mass \( m \) subject to a symmetric linear potential. The potential energy is given by:
\[ V(x) = k|x| = \begin{cases} kx & x > 0 \\ -kx & x < 0 \end{cases} \]
The time-independent Schrödinger Equation (TISE) is:
\[ -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + k|x|\psi = E\psi \]
2. Exact Solution: Derivation of the Airy Equation
Since the potential is symmetric, the wavefunctions will have definite parity (Even or Odd). We can solve for the region \( x > 0 \) and apply boundary conditions at \( x = 0 \).
Step A: Rearranging the Schrödinger Equation
For \( x > 0 \), \( V(x) = kx \). The equation becomes:
\[ -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + kx\psi = E\psi \]
Rearrange to isolate the second derivative:
\[ \frac{d^2\psi}{dx^2} = \frac{2m}{\hbar^2}(kx - E)\psi \]
Step B: Variable Substitution (Scaling)
We aim to transform this into a dimensionless form. First, we shift the origin to the classical turning point.
1. Coordinate Shift:
Let \( \xi = x - \frac{E}{k} \).
Then \( d\xi = dx \) and \( \frac{d^2}{dx^2} = \frac{d^2}{d\xi^2} \).
The term \( (kx - E) \) simplifies to:
\[ k(x - E/k) = k\xi \]
Substituting this back:
\[ \frac{d^2\psi}{d\xi^2} = \frac{2mk}{\hbar^2} \xi \psi \quad \dots(\text{Eq. 1}) \]
2. Dimensionless Scaling:
We define a new dimensionless variable \( z \) by multiplying \( \xi \) by a scaling factor \( \alpha \):
\[ z = \alpha \xi \]
From chain rule differentiation:
\[ \frac{d}{d\xi} = \frac{dz}{d\xi}\frac{d}{dz} = \alpha \frac{d}{dz} \]
\[ \frac{d^2}{d\xi^2} = \alpha^2 \frac{d^2}{dz^2} \]
Substituting these into Eq. 1:
\[ \alpha^2 \frac{d^2\psi}{dz^2} = \frac{2mk}{\hbar^2} \left( \frac{z}{\alpha} \right) \psi \]
Divide the entire equation by \( \alpha^2 \):
\[ \frac{d^2\psi}{dz^2} = \left[ \frac{2mk}{\hbar^2 \alpha^3} \right] z \psi \]
3. Determining \( \alpha \):
To obtain the standard Airy equation form \( \psi'' = z\psi \), the term in the brackets must equal 1.
\[ \frac{2mk}{\hbar^2 \alpha^3} = 1 \implies \alpha^3 = \frac{2mk}{\hbar^2} \] \[ \alpha = \left( \frac{2mk}{\hbar^2} \right)^{1/3} \]
This yields the standard Airy Differential Equation:
\[ \frac{d^2\psi}{dz^2} - z\psi = 0 \]
Step C: Solutions and Boundary Conditions
The general mathematical solution is:
\[ \psi(z) = C_1 \text{Ai}(z) + C_2 \text{Bi}(z) \]
As \( x \to \infty \), \( z \to \infty \). The Airy function of the second kind, \( \text{Bi}(z) \), diverges exponentially as \( z \to \infty \). For a physical wavefunction to be normalizable, we must set \( C_2 = 0 \).
The solution is: \[ \psi(x) \propto \text{Ai}(z) = \text{Ai}\left( \alpha(x - E/k) \right) \]
Figure 2: The behavior of Ai(z) (decaying) and Bi(z) (diverging) for z > 0.
Step D: Eigenvalue Conditions (Parity)
The allowed energies depend on the boundary condition at the origin \( x=0 \). At this point, the argument of the Airy function is \( z_0 = \alpha(0 - E/k) = -\alpha E/k \).
- Odd Parity Solutions (\( \psi(-x) = -\psi(x) \)):
Must satisfy \( \psi(0) = 0 \).
Therefore: \( \text{Ai}\left( -\alpha \frac{E}{k} \right) = 0 \). The energies correspond to the zeros of the Airy function. - Even Parity Solutions (\( \psi(-x) = \psi(x) \)):
Must satisfy slope continuity \( \psi'(0) = 0 \).
Therefore: \( \text{Ai}'\left( -\alpha \frac{E}{k} \right) = 0 \). The energies correspond to the zeros of the derivative of the Airy function.
3. WKB Approximation: Full Derivation
The WKB method provides an approximate analytical formula for the energy levels for high quantum numbers \( n \).
Step A: The Quantization Integral
The Wilson-Sommerfeld quantization condition for a potential well with two turning points is:
\[ \int_{x_1}^{x_2} p(x) dx = \left(n + \frac{1}{2}\right)\pi\hbar \]
The classical turning points occur where \( E = V(x) \).
For \( V(x) = k|x| \), the turning points are \( x = \pm E/k \).
Using the symmetry of the potential, we integrate from 0 to \( E/k \) and multiply by 2:
\[ 2 \int_{0}^{E/k} \sqrt{2m(E - kx)} \, dx = \left(n + \frac{1}{2}\right)\pi\hbar \]
Step B: Evaluating the Integral
Let us solve the integral \( I = \int_{0}^{E/k} \sqrt{E - kx} \, dx \). We use u-substitution.
Substitution Details:
Let \( u = E - kx \).
Then \( du = -k \, dx \implies dx = -\frac{du}{k} \).
Limits:
When \( x = 0 \), \( u = E - 0 = E \).
When \( x = E/k \), \( u = E - k(E/k) = 0 \).
Substitute into the integral:
\[ I = \int_{E}^{0} \sqrt{u} \left( -\frac{du}{k} \right) \]
Reverse limits (removes the negative sign):
\[ I = \frac{1}{k} \int_{0}^{E} u^{1/2} \, du \]
Integrate using power rule \( \int u^n du = \frac{u^{n+1}}{n+1} \):
\[ I = \frac{1}{k} \left[ \frac{u^{3/2}}{3/2} \right]_{0}^{E} = \frac{2}{3k} E^{3/2} \]
Step C: Solving for Energy \( E_n \)
Substitute the result of the integral \( I \) back into the quantization equation:
\[ 2\sqrt{2m} \underbrace{\left( \frac{2}{3k} E^{3/2} \right)}_{\text{Integral Result}} = \left(n + \frac{1}{2}\right)\pi\hbar \]
Combine constants:
\[ \frac{4\sqrt{2m}}{3k} E^{3/2} = \left(n + \frac{1}{2}\right)\pi\hbar \]
Rearrange to isolate \( E^{3/2} \):
\[ E^{3/2} = \frac{3k\pi\hbar}{4\sqrt{2m}} \left(n + \frac{1}{2}\right) \]
Raise both sides to the power of \( 2/3 \) to get the final expression:
Final WKB Energy Spectrum Formula:
\[ E_n = \left[ \frac{3\pi\hbar k}{4\sqrt{2m}} \left(n + \frac{1}{2}\right) \right]^{2/3} \]
Figure 3: Comparison of WKB approximation (dashed) vs exact Airy function solution (solid).
4. Conclusion
The derivation leads to two significant physical insights:
- Energy Scaling: The energy levels scale as \( E_n \propto n^{2/3} \). This is distinct from the Hydrogen atom (\( \propto -1/n^2 \)) or the Harmonic Oscillator (\( \propto n \)).
- Level Spacing: As \( n \) increases, the difference between adjacent energy levels decreases (\( \Delta E \propto n^{-1/3} \)). This confirms that the potential well widens significantly at higher energies, reducing the "confinement pressure" on the states.