Quantum Tunneling Probability & Barrier Calculator: Free Quantum Tunneling Probability & Barrier Calculator tool: calculate quantum tunneling probability T(E), evanescent decay constant kappa, transmission.
1. The Quantum Mechanical Barrier Penetration Phenomenon
In classical Newtonian mechanics, energy conservation is absolute: a particle of mass $m$ possessing total kinetic energy $E$ encountering a potential energy barrier of height $V_0$ must experience a hard physical rebound if $E < V_0$. Inside the barrier, kinetic energy would be $E_k = E - V_0 < 0$, implying an imaginary velocity ($v = sqrt{2(E - V_0)/m} in mathbb{C}$), which is strictly forbidden in classical kinematics.
In 1926, Erwin Schrödinger formulated the wave equation governing quantum particles, and in 1928, physicists George Gamow, Ronald Gurney, and Edward Condon demonstrated that microscopic matter behaves as a probability wave amplitude $psi(x)$. The One-Dimensional Time-Independent Schrödinger Equation states:
For a finite rectangular potential barrier of width $a$ and height $V_0$:
In Region II where $E < V_0$, the Schrödinger equation rearranges to:
Rather than oscillating sinusoidally, the spatial solution inside the barrier is a linear superposition of decaying and growing real exponentials: $psi_{text{II}}(x) = B_1 e^{-kappa x} + B_2 e^{+kappa x}$. If the barrier width $a$ is on the order of a few nanometers or less, the exponential tail does not decay completely to zero at $x = a$. A non-vanishing probability amplitude penetrates through, emerging in Region III as an attenuated propagating sinusoidal matter wave!
2. Exact Analytical Derivation of the Transmission Coefficient T(E)
By enforcing continuity of both the wavefunction $psi(x)$ and its first spatial derivative $frac{dpsi}{dx}$ at the boundaries $x = 0$ and $x = a$:
psi_{text{II}}(a) = psi_{text{III}}(a), quad left.frac{dpsi_{text{II}}}{dx}right|_{x=a} = left.frac{dpsi_{text{III}}}{dx}right|_{x=a}
Solving the system of boundary equations yields the Exact Transmission Coefficient ($T$) for $E < V_0$:
When the barrier is sufficiently thick or high such that $kappa a gg 1$, $sinh(kappa a) approx frac{1}{2} e^{kappa a} implies sinh^2(kappa a) approx frac{1}{4} e^{2kappa a}$. Substituting this yields the classical Gamow Approximation:
For above-barrier propagation ($E > V_0$), the transmission exhibits quantum interference fringes (Ramsauer-Townsend resonances):
Whenever $k_2 a = npi$ ($n = 1, 2, 3 dots$), $sin^2(k_2 a) = 0$, resulting in $100%$ perfect transmission ($T = 1.0$) with zero quantum reflection!
3. Technological & Astrophysical Applications of Quantum Tunneling
Far from being an esoteric theoretical curiosity, quantum tunneling underpins critical modern electronic technologies and fundamental astrophysical mechanisms:
| Application / Phenomenon | Typical Barrier Width ($a$) | Potential Barrier ($V_0$) | Governing Quantum Particle | Physical Impact |
|---|---|---|---|---|
| Flash Memory & NAND Gate Writing | 1.0 – 3.0 nm | ~3.1 eV ($text{SiO}_2$ gate oxide) | Conduction Electron | Fowler-Nordheim tunneling traps charge on floating gates. |
| Scanning Tunneling Microscope (STM) | 0.4 – 1.0 nm | ~4.5 eV (Work function gap) | Electron | Sub-angstrom atomic scale topography imaging. |
| Nuclear Alpha Decay ($alpha$-Decay) | ~20 – 40 fm | ~25 – 35 MeV (Coulomb barrier) | Alpha Particle ($^4text{He}$ nucleus) | Explains uranium/radium half-lives spanning $10^{-7}text{ s}$ to $10^{10}text{ yrs}$. |
| Solar Core Proton-Proton Chain | ~1 fm | ~1,000 keV (Coulomb repulsion) | Proton ($^1text{H}$) | Enables stellar hydrogen burning at modest 1.5 keV core temperatures. |
| Superconducting Josephson Junction | 1.0 – 2.0 nm | Insulating oxide barrier | Cooper Pairs ($2e^-$) | SQUID magnetometers and superconducting quantum qubits. |
4. Worked Quantum Mechanical Calculation: Flash Memory Gate Oxide Tunneling
Calculate the quantum tunneling probability of a conduction electron tunneling through a thin silicon dioxide ($text{SiO}_2$) dielectric barrier in a high-density 3D NAND flash memory cell:
- Particle: Electron, rest mass $m_e = 9.10938 times 10^{-31}text{ kg}$, reduced Planck constant $hbar = 1.05457 times 10^{-34}text{ J}cdottext{s}$.
- Barrier Potential Height: $V_0 = 10.0text{ eV} = 10.0 times 1.60218 times 10^{-19}text{ J} = 1.60218 times 10^{-18}text{ J}$.
- Electron Kinetic Energy: $E = 5.0text{ eV} = 5.0 times 1.60218 times 10^{-19}text{ J} = 8.0109 times 10^{-19}text{ J}$.
- Barrier Thickness: $a = 0.50text{ nm} = 5.0 times 10^{-10}text{ m}$.
Step A: Energy Deficit & Evanescent Decay Constant ($kappa$)
kappa = frac{sqrt{2 m_e (V_0 - E)}}{hbar} = frac{sqrt{2 times (9.10938 times 10^{-31}) times (8.0109 times 10^{-19})}}{1.05457 times 10^{-34}}
= frac{sqrt{1.45949 times 10^{-48}}}{1.05457 times 10^{-34}} = frac{1.20809 times 10^{-24}}{1.05457 times 10^{-34}} approx 1.14558 times 10^{10}text{ m}^{-1}
The characteristic penetration attenuation depth is $delta = 1/kappa = 1 / (1.1456 times 10^{10}) approx 8.73 times 10^{-11}text{ m} = 0.0873text{ nm} approx 0.87text{ Å}$.
Step B: Dimensionless Barrier Product ($kappa a$)
2 kappa a = 2 times 5.7279 = 11.4558
Step C: Exact Transmission Coefficient ($T$)
sinh^2(kappa a) = (153.66)^2 approx 23,611.4
frac{V_0^2}{4E(V_0 - E)} = frac{10^2}{4 times 5 times (10 - 5)} = frac{100}{4 times 25} = frac{100}{100} = 1.000
T = frac{1}{1 + 1.000 times sinh^2(kappa a)} = frac{1}{1 + 23,611.4} = frac{1}{23,612.4} approx 4.235 times 10^{-5}
The quantum transmission probability is $T approx 4.24 times 10^{-5}$ ($0.00424%$). For every $1,000,000$ electrons striking the barrier, approximately $42$ particles tunnel directly through the classically impenetrable barrier.
Step D: High-Barrier Exponential Approximation Check
= 4.0 times (1.0588 times 10^{-5}) = 4.235 times 10^{-5}
The Gamow approximation matches the exact hyperbolic formula to four significant digits, confirming its accuracy when $kappa a > 3$.
5. The Wavefunction Amplitude Profile Across Barrier Interfaces
Examining the spatial probability density $|psi(x)|^2$ provides profound intuition:
- Region I ($x < 0$): The incident wave $e^{ikx}$ interferes with the reflected wave $R e^{-ikx}$, forming a partial standing wave with pronounced interference peaks and nodes.
- Region II ($0 le x le a$): The wavefunction decays rapidly without spatial nodes. No net energy is consumed by the barrier; it acts purely as a reactive spatial attenuator.
- Region III ($x > a$): The transmitted wave propagates forward with a reduced amplitude $|T|^2$, but possessing the exact same wavelength ($lambda = h / sqrt{2mE}$) and momentum as the original incident particle!