3D Quantum Black Hole in Spherical Box
Modeling a microscopic black hole as a quantum particle in a 3D spherical infinite well with proper angular momentum quantization
Key Advancement: Realistic 3D Geometry
Moving from 1D to 3D spherical geometry properly accounts for:
1. Spherical symmetry of actual black holes
2. Angular momentum quantization (quantum numbers ℓ and m)
3. Radial probability distributions that match Schwarzschild geometry
4. Realistic tunneling probabilities through spherical barriers
1D vs 3D Model Comparison
Simplified Model: Particle confined to line segment 0 < x < L
Wavefunction: ψₙ(x) = √(2/L) sin(nπx/L)
Energy: Eₙ = n²π²ħ²/(2mL²)
Single quantum number: n = 1, 2, 3, ...
Limitations: No angular dependence, unrealistic for spherical black holes
Realistic Geometry: Particle confined to sphere of radius R
Wavefunction: ψₙℓₘ(r,θ,φ) = Rₙℓ(r)Yℓₘ(θ,φ)
Energy: Eₙℓ = (ħ²/2mR²)αₙℓ² where αₙℓ are zeros of Bessel functions
Three quantum numbers: n (radial), ℓ (angular), m (magnetic)
Advantages: Proper spherical symmetry, angular momentum quantization
Mathematical Formulation: 3D Schrödinger Equation
In spherical coordinates, the Laplacian operator is:
For a spherical infinite well of radius R (black hole radius):
Assume solution factorizes: ψ(r,θ,φ) = R(r)Y(θ,φ)
Radial and angular parts separate with separation constant ℓ(ℓ+1):
where Λ is the angular part of Laplacian.
The angular equation gives spherical harmonics Yℓₘ(θ,φ):
where ℓ = 0, 1, 2, ... and m = -ℓ, -ℓ+1, ..., ℓ-1, ℓ
For V(r) = 0 inside well, radial equation becomes:
Solution: Rₙℓ(r) = A jℓ(kr) where jℓ(x) are spherical Bessel functions.
Wavefunction must vanish at boundary: ψ(R,θ,φ) = 0 ⇒ jℓ(kR) = 0
Define αₙℓ as the n-th zero of jℓ(x): jℓ(αₙℓ) = 0
Then: kₙℓ = αₙℓ/R and Eₙℓ = (ħ²αₙℓ²)/(2mR²)
Radius R = Schwarzschild Radius
Probability Density
Spherical Harmonics Yℓₘ(θ,φ)
Quantum Numbers and Energy Levels
n = 1, 2, 3, ...
Number of radial nodes
ℓ = 0, 1, 2, ..., n-1
Orbital angular momentum
m = -ℓ, ..., ℓ
z-component of angular momentum
| State (n,ℓ) | αₙℓ (Bessel zero) | Energy Eₙℓ/E₁₀ | Angular Momentum | Degeneracy (2ℓ+1) |
|---|---|---|---|---|
| Ground state (1,0) | π ≈ 3.1416 | 1.000 | 0 (s-orbital) | 1 |
| First excited (1,1) | 4.4934 | 2.046 | √2ħ (p-orbital) | 3 |
| (1,2) | 5.7635 | 3.363 | √6ħ (d-orbital) | 5 |
| (2,0) | 6.2832 (2π) | 4.000 | 0 | 1 |
| (1,3) | 6.9879 | 4.946 | √12ħ (f-orbital) | 7 |
Note: For our black hole model, the ground state (1,0) corresponds to the minimum black hole with zero angular momentum (Schwarzschild black hole). Higher ℓ states correspond to rotating black holes (Kerr black holes).
Spherical Solutions for Different States
Spherically symmetric (s-orbital)
Maximum probability at r=0 (center)
Zero angular momentum
Angular dependence: Y₁₀ ∝ cosθ
Probability maximum off-center
Angular momentum ħ
More complex angular pattern
Angular momentum √6ħ
Higher energy state
One radial node at r=R/2
Lower probability at center
Higher energy (4× ground state)
3D Tunneling Through Spherical Barrier
Replace infinite wall with finite potential barrier:
where V₀ is gravitational potential barrier height, a is barrier width.
Inside (r < R): ψ₁(r) = A jℓ(k₁r) where k₁ = √(2mE)/ħ
Barrier (R ≤ r ≤ R+a): ψ₂(r) = B hℓ⁽¹⁾(iκr) + C hℓ⁽²⁾(iκr)
where κ = √[2m(V₀-E)]/ħ and hℓ are spherical Hankel functions
Outside (r > R+a): ψ₃(r) = D hℓ⁽¹⁾(k₁r) (outgoing wave only)
For ℓ=0 (s-wave tunneling, most probable):
For constant V₀ and ℓ=0: T ≈ exp(-2κa)
For ℓ>0: Additional centrifugal barrier reduces tunneling probability.
For black hole, V₀ ≈ mc² at horizon (rest mass energy)
Barrier width a ≈ few × Planck length for minimum black hole
Tunneling probability becomes extremely small but non-zero.
Python Implementation: 3D Quantum Black Hole
import numpy as np
from scipy.special import spherical_jn, spherical_yn, sph_harm
from scipy.optimize import root
class QuantumBlackHole3D:
"""3D quantum black hole in spherical box model"""
def __init__(self, M, G=6.67430e-11, c=2.99792458e8, hbar=1.054571817e-34):
self.M = M # Black hole mass (kg)
self.G = G
self.c = c
self.hbar = hbar
# Schwarzschild radius
self.R = 2 * G * M / c**2
# For particle in box, we use effective mass (could be M or reduced mass)
self.m = M # Simplified: black hole as single quantum entity
# Bessel function zeros (αₙℓ) for n=1
self.alpha_zeros = {
(1,0): np.pi, # j₀(π)=0
(1,1): 4.4934094579, # j₁(x)=0 first zero
(1,2): 5.7634591969, # j₂(x)=0 first zero
(1,3): 6.9879320005, # j₃(x)=0 first zero
(2,0): 2*np.pi, # j₀(2π)=0
(2,1): 7.7252518369, # j₁(x)=0 second zero
}
def energy_level(self, n, l):
"""Energy of state (n,l) in spherical infinite well"""
if (n,l) in self.alpha_zeros:
alpha = self.alpha_zeros[(n,l)]
else:
# Find zero of spherical Bessel function j_l(x)
alpha = self.find_bessel_zero(n, l)
return (self.hbar**2 * alpha**2) / (2 * self.m * self.R**2)
def find_bessel_zero(self, n, l):
"""Find n-th zero of spherical Bessel function j_l(x)"""
# Define function whose root we want: j_l(x) = 0
def func(x):
return spherical_jn(l, x)
# Initial guess based on asymptotic formula
# Zeros of j_l(x) ~ π(n + l/2) for large n
guess = np.pi * (n + l/2)
# Use root finding (simplified - in practice need careful handling)
# This is a simplified version
result = root(func, guess)
return result.x[0]
def radial_wavefunction(self, n, l, r_points=1000):
"""Calculate radial wavefunction R_nl(r)"""
alpha = self.alpha_zeros.get((n,l), self.find_bessel_zero(n, l))
k = alpha / self.R
r = np.linspace(0, self.R, r_points)
R = spherical_jn(l, k * r)
# Normalize: ∫|R(r)|² r² dr = 1
norm = np.trapz(R**2 * r**2, r)
R_normalized = R / np.sqrt(norm)
return r, R_normalized
def probability_density(self, n, l, m, theta=0, phi=0, r_points=1000):
"""Calculate full probability density |ψ_nlm(r,θ,φ)|²"""
r, R_nl = self.radial_wavefunction(n, l, r_points)
# Spherical harmonic
Y_lm = sph_harm(m, l, phi, theta)
# Full wavefunction ψ_nlm(r,θ,φ) = R_nl(r) * Y_lm(θ,φ)
# Probability density integrated over angles gives radial probability
psi = R_nl * Y_lm
return r, np.abs(psi)**2
def tunneling_probability(self, n, l, V0, a):
"""
Estimate tunneling probability through finite barrier
V0: Barrier height (J)
a: Barrier width (m)
"""
E = self.energy_level(n, l)
if E >= V0:
return 1.0 # Above barrier, not tunneling
# For ℓ=0, simple WKB approximation
kappa = np.sqrt(2 * self.m * (V0 - E)) / self.hbar
# For ℓ>0, additional centrifugal barrier
centrifugal = self.hbar**2 * l*(l+1) / (2 * self.m * self.R**2)
kappa_eff = np.sqrt(2 * self.m * (V0 + centrifugal - E)) / self.hbar
# Transmission probability (simplified)
T = np.exp(-2 * kappa_eff * a)
return T
def hawking_temperature(self):
"""Hawking temperature for this black hole"""
return self.hbar * self.c**3 / (8 * np.pi * self.G * self.M * 1.380649e-23)
def evaporation_time(self):
"""Time for complete evaporation via Hawking radiation"""
return (5120 * np.pi * self.G**2 * self.M**3) / (self.hbar * self.c**4)
# Example calculation for minimum black hole
M_min = 6.5e-9 # kg (from previous calculation)
bh = QuantumBlackHole3D(M_min)
print("3D Quantum Black Hole Model")
print(f"Mass: {M_min:.2e} kg")
print(f"Schwarzschild radius: {bh.R:.2e} m")
print(f"\nEnergy levels (in Joules):")
for (n,l) in [(1,0), (1,1), (1,2), (2,0)]:
E = bh.energy_level(n, l)
print(f" State (n={n}, l={l}): E = {E:.2e} J = {E/1.602e-19:.2e} eV")
print(f"\nGround state energy: {bh.energy_level(1,0):.2e} J")
print(f"Mass energy (Mc²): {M_min * (2.9979e8)**2:.2e} J")
# Calculate tunneling probability
V0 = M_min * bh.c**2 # Rest mass energy as barrier height
a = 1.616e-35 # Planck length as barrier width
T_tunnel = bh.tunneling_probability(1, 0, V0, a)
print(f"\nTunneling probability (ℓ=0, ground state): {T_tunnel:.2e}")
# Compare with Hawking radiation rate
T_H = bh.hawking_temperature()
print(f"Hawking temperature: {T_H:.2e} K")
print(f"Evaporation time: {bh.evaporation_time():.2e} s")
# Generate radial probability distribution
import matplotlib.pyplot as plt
r, prob = bh.probability_density(1, 0, 0, theta=0, phi=0)
plt.figure(figsize=(10, 6))
plt.plot(r/bh.R, prob**2 * r**2, 'b-', linewidth=2, label='Radial probability density')
plt.xlabel('r / R (normalized distance)')
plt.ylabel('Probability density × r²')
plt.title('3D Quantum Black Hole: Ground State (n=1, ℓ=0)')
plt.grid(True, alpha=0.3)
plt.legend()
plt.show()
Physical Interpretation & Implications
| Aspect | 1D Model | 3D Spherical Model | Physical Significance |
|---|---|---|---|
| Ground State | n=1: ψ₁(x)=√(2/L)sin(πx/L) | n=1, ℓ=0: ψ₁₀₀∝sin(πr/R)/r | Spherically symmetric minimum black hole |
| Angular Momentum | Not accounted for | Quantized: L²=ℓ(ℓ+1)ħ² | ℓ>0 states = rotating (Kerr) black holes |
| Degeneracy | 1 (non-degenerate) | 2ℓ+1 (m=-ℓ,...,ℓ) | Multiple states with same energy but different orientation |
| Tunneling | Simple exponential decay | Angular momentum barrier reduces tunneling for ℓ>0 | s-waves (ℓ=0) tunnel most easily |
| Black Hole Type | Schwarzschild only | ℓ=0: Schwarzschild, ℓ>0: Kerr-like | More realistic classification of quantum black holes |
1. Minimum Black Hole is Spherically Symmetric: The ground state (n=1, ℓ=0) has no angular momentum, corresponding to a non-rotating Schwarzschild black hole.
2. Rotating Quantum Black Holes: States with ℓ>0 represent microscopic rotating black holes with quantized angular momentum.
3. Higher ℓ States Tunnel Less: The centrifugal barrier ℓ(ℓ+1)ħ²/(2mr²) makes tunneling less probable for states with angular momentum.
4. Realistic Probability Distribution: The 3D radial probability |R(r)|²r² peaks at specific radii, unlike 1D model's simple sine squared.
Conclusion: Advantages of 3D Model
The 3D spherical model properly captures the spherical symmetry of actual black holes, unlike the 1D simplification. Black holes are fundamentally 3D objects, and the spherical well is the natural quantum analog.
The 3D model naturally incorporates quantized angular momentum through quantum numbers ℓ and m. This allows us to distinguish between non-rotating (ℓ=0) and rotating (ℓ>0) quantum black holes, corresponding to Schwarzschild vs. Kerr black holes in classical GR.
Tunneling probabilities in 3D account for centrifugal barriers that reduce tunneling for states with angular momentum. The s-wave (ℓ=0) tunneling is most probable, which aligns with expectations from quantum field theory in curved spacetime.
This 3D model can be extended to include:
- Relativistic corrections (Klein-Gordon or Dirac equation instead of Schrödinger)
- Finite temperature effects (black hole thermodynamics)
- Quantum field theory in the curved background
- Backreaction of emitted particles on spacetime
Final Insight: While the 1D particle-in-a-box model provides valuable conceptual insights, the 3D spherical model is essential for realistic quantum descriptions of black holes. It reveals that the minimum quantum black hole is a spherically symmetric, zero-angular-momentum object (ℓ=0 ground state), and that rotating quantum black holes correspond to excited states with ℓ>0. The tunneling dynamics are more complex but more physically accurate in 3D, with s-wave tunneling dominating the evaporation process.
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