Part 8, Chapter 6

Quantum Plasmas

Degenerate plasmas, Bohm potential, and quantum dispersion

6.1 When Does a Plasma Become Quantum?

Classical plasma physics assumes that the de Broglie wavelength of the particles is much smaller than the interparticle spacing. When this assumption breaks down, quantum effects become important. The quantum degeneracy parameter quantifies this:

$$\chi = n_e \lambda_{dB}^3$$

where the thermal de Broglie wavelength is:

$$\lambda_{dB} = \frac{h}{\sqrt{2\pi m_e T_e}}$$

When $\chi \ll 1$, the plasma is classical (Maxwell-Boltzmann statistics). When $\chi \gtrsim 1$, the electron wavefunctions overlap and Fermi-Dirac statistics must be used. The electrons become degenerate, with the Fermi energy replacing the thermal energy as the relevant energy scale:

$$E_F = \frac{\hbar^2}{2m_e}(3\pi^2 n_e)^{2/3}$$

Quantum plasmas arise in white dwarf stars ($n_e \sim 10^{30}$ cm^-3), neutron star crusts, giant planet interiors, and semiconductor quantum wells. They also appear in intense laser-solid interactions where compression creates extreme densities.

6.2 The Quantum Bohm Potential

In the quantum hydrodynamic (QHD) formulation, the Schrodinger equation for the electron fluid can be rewritten in the Madelung form by decomposing the wavefunction as$\psi = \sqrt{n_e}\, e^{iS/\hbar}$. This yields fluid-like continuity and momentum equations, with an additional quantum force arising from the Bohm potential:

$$Q = -\frac{\hbar^2}{2m_e}\frac{\nabla^2\sqrt{n_e}}{\sqrt{n_e}}$$

The Bohm potential can be expanded as:

$$Q = -\frac{\hbar^2}{4m_e}\left(\frac{\nabla^2 n_e}{n_e} - \frac{|\nabla n_e|^2}{2n_e^2}\right)$$

The quantum electron momentum equation becomes:

$$m_e n_e \frac{d\mathbf{v}_e}{dt} = -\nabla p_e + en_e\mathbf{E} + n_e\nabla Q$$

The Bohm potential introduces a new dispersive length scale. For density perturbations at wavenumber $k$, the Bohm term contributes an effective pressure $\sim \hbar^2 k^2 n_e/(4m_e)$. This becomes important when the wavelength approaches the de Broglie wavelength or when the density has sharp spatial gradients.

6.3 Wigner Function Formalism

The Wigner function provides a phase-space representation of quantum mechanics that is the natural quantum analog of the classical distribution function. It is defined as:

$$f_W(\mathbf{r}, \mathbf{p}, t) = \frac{1}{(\pi\hbar)^3}\int \psi^*\!\left(\mathbf{r}+\mathbf{s}\right)\psi\!\left(\mathbf{r}-\mathbf{s}\right) e^{2i\mathbf{p}\cdot\mathbf{s}/\hbar}\, d^3s$$

The Wigner function evolves according to the quantum Vlasov (Wigner-Moyal) equation:

$$\frac{\partial f_W}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla_r f_W + \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}\left(\frac{\hbar}{2}\right)^{2n} \nabla_r^{2n+1}V \cdot \nabla_p^{2n+1} f_W = 0$$

The $n=0$ term gives the classical Vlasov equation. Higher-order terms encode quantum corrections in powers of $\hbar^2$. Unlike a classical distribution, the Wigner function can take negative values, reflecting quantum interference. However, its marginals (integrals over position or momentum) are always non-negative probability distributions.

6.4 Quantum Langmuir Dispersion

Including the Bohm potential and the Fermi pressure in the electron momentum equation modifies the Langmuir wave dispersion. For a zero-temperature degenerate electron gas, the dispersion relation becomes:

$$\omega^2 = \omega_{pe}^2 + 3k^2 v_{Fe}^2 + \frac{\hbar^2 k^4}{4m_e^2}$$

where $v_{Fe} = \sqrt{2E_F/m_e}$ is the Fermi velocity. Compare this to the classical (Bohm-Gross) dispersion:

$$\omega^2 = \omega_{pe}^2 + 3k^2 v_{te}^2 \quad \text{(classical)}$$

The quantum correction $\hbar^2 k^4/(4m_e^2)$ is a purely dispersive term that becomes dominant at very short wavelengths (comparable to the de Broglie wavelength). The crossover wavenumber where quantum effects become comparable to thermal effects is:

$$k_Q \sim \frac{2m_e v_{Fe}}{\hbar} = 2k_F$$

where $k_F = (3\pi^2 n_e)^{1/3}$ is the Fermi wavenumber.

Interactive Simulation

Classical vs Quantum Langmuir Dispersion and Bohm Potential

Python

script.py109 lines

import numpy as np

# Constants
hbar = 1.055e-34
m_e = 9.109e-31
e = 1.602e-19
eps0 = 8.854e-12
h = 6.626e-34
k_B = 1.381e-23

print("=== Quantum Degeneracy Parameter ===")
print()

# Different plasma environments
environments = [
    ("Tokamak core", 1e20, 1e4),          # n_e (m^-3), T_e (eV)
    ("Solar corona", 1e15, 100),
    ("Metal (Cu)", 8.5e28, 7.0),           # Fermi energy ~7 eV
    ("White dwarf", 1e36, 1e4),
    ("Laser-compressed", 1e32, 100),
    ("Semiconductor (n-GaAs)", 1e24, 0.026),
]

print(f"{'Environment':>22} {'n_e (m^-3)':>12} {'T_e (eV)':>10} {'lambda_dB (m)':>14} {'chi':>12} {'Regime':>12}")
print("-" * 86)

for name, n_e, T_eV in environments:
    T_e = T_eV * e  # Joules
    lambda_dB = h / np.sqrt(2 * np.pi * m_e * T_e)
    chi = n_e * lambda_dB**3
    regime = "QUANTUM" if chi > 1 else "classical"
    print(f"{name:>22} {n_e:12.1e} {T_eV:10.1f} {lambda_dB:14.2e} {chi:12.2e} {regime:>12}")

# ==============================
# Quantum vs Classical Langmuir Dispersion
# ==============================
print()
print("=== Quantum vs Classical Langmuir Dispersion ===")
print()

# Use white dwarf parameters
n_e_wd = 1e36    # m^-3
T_e_wd = 1e4 * e  # J

omega_pe = np.sqrt(n_e_wd * e**2 / (eps0 * m_e))
v_te = np.sqrt(T_e_wd / m_e)
E_F = hbar**2 / (2 * m_e) * (3 * np.pi**2 * n_e_wd)**(2/3)
v_Fe = np.sqrt(2 * E_F / m_e)
k_F = (3 * np.pi**2 * n_e_wd)**(1/3)

print(f"White dwarf parameters:")
print(f"  n_e       = {n_e_wd:.1e} m^-3")
print(f"  omega_pe  = {omega_pe:.3e} rad/s")
print(f"  v_te      = {v_te:.3e} m/s")
print(f"  E_F       = {E_F/e:.1f} eV")
print(f"  v_Fe      = {v_Fe:.3e} m/s")
print(f"  k_F       = {k_F:.3e} m^-1")
print(f"  T_e/E_F   = {T_e_wd/E_F:.4f} (degenerate: T << E_F)")
print()

# Dispersion comparison
k_vals = np.logspace(8, np.log10(3*k_F), 60)

omega_classical = np.sqrt(omega_pe**2 + 3 * k_vals**2 * v_te**2)
omega_quantum = np.sqrt(omega_pe**2 + 3 * k_vals**2 * v_Fe**2 + hbar**2 * k_vals**4 / (4 * m_e**2))
omega_bohm_only = np.sqrt(omega_pe**2 + hbar**2 * k_vals**4 / (4 * m_e**2))

print(f"{'k/k_F':>8} {'w_cl/w_pe':>12} {'w_qu/w_pe':>12} {'w_bohm/w_pe':>12} {'Ratio qu/cl':>12}")
print("-" * 60)
for i in range(0, len(k_vals), 6):
    k = k_vals[i]
    print(f"{k/k_F:8.3f} {omega_classical[i]/omega_pe:12.4f} {omega_quantum[i]/omega_pe:12.4f} {omega_bohm_only[i]/omega_pe:12.4f} {omega_quantum[i]/omega_classical[i]:12.4f}")

# ==============================
# Bohm Potential for a Density Perturbation
# ==============================
print()
print("=== Bohm Potential Profile ===")
print()

# Consider a density perturbation: n_e(x) = n0 * (1 + delta * cos(k*x))
n0 = 1e36
delta = 0.1   # Perturbation amplitude

# Different wavelengths
wavelengths_nm = [0.01, 0.05, 0.1, 0.5, 1.0]

print(f"Background density: {n0:.0e} m^-3, perturbation delta = {delta}")
print(f"Bohm potential: Q = -(hbar^2/2m_e) * (d^2 sqrt(n)/dx^2) / sqrt(n)")
print()
print(f"{'lambda (nm)':>12} {'k (m^-1)':>12} {'Q_max (eV)':>12} {'Q/E_F':>10} {'k/k_F':>10}")
print("-" * 60)

for lam_nm in wavelengths_nm:
    lam = lam_nm * 1e-9
    k = 2 * np.pi / lam

# For small delta: sqrt(n) ~ sqrt(n0)*(1 + delta/2 * cos(kx))
    # d^2(sqrt(n))/dx^2 ~ -sqrt(n0) * delta/2 * k^2 * cos(kx)
    # Q ~ hbar^2*k^2*delta/(4*m_e) * cos(kx)
    Q_max = hbar**2 * k**2 * delta / (4 * m_e)
    Q_eV = Q_max / e

print(f"{lam_nm:12.3f} {k:12.2e} {Q_eV:12.4f} {Q_max/E_F:10.4f} {k/k_F:10.4f}")

print()
print("At k ~ k_F, the Bohm potential becomes comparable to the Fermi energy,")
print("and quantum diffraction effects fundamentally alter wave propagation.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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