Quantum Tunneling through Potential Barriers: Difference between revisions

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'''Page Claimed By: Joshua Kim Fall '25'''
'''Page Claimed By: Haripratiik Arunkumar Malarkodi Spring '26'''


'''Quantum tunneling''' is a fundamental effect in quantum mechanics in which a particle has a nonzero probability of appearing on the far side of a potential barrier, even when its classical energy is insufficient to cross that barrier. This behavior arises from the wave-like nature of matter and is described mathematically through the Schrodinger equation and related quantum principles, such as the Heisenberg uncertainty principle.
'''Quantum tunneling''' is a fundamental effect in quantum mechanics in which a particle has a nonzero probability of passing through a potential barrier even when its classical energy is too small to go over the barrier. In classical mechanics, this would be impossible. In quantum mechanics, particles are described by wavefunctions, and those wavefunctions can extend into regions that would be classically forbidden.
 
Quantum tunneling is important because it explains physical processes that classical physics cannot explain, including alpha decay, nuclear fusion in stars, scanning tunneling microscopy, and tunneling effects in semiconductor devices.


==Overview==
==Overview==
In classical physics, a particle with energy lower than a barrier height cannot enter the barrier at all. Quantum mechanics, however, treats particles as wavefunctions rather than solid objects. These wavefunctions can extend into and beyond regions that would be forbidden in classical mechanics, giving rise to a small--but finite--chance that the particle will emerge on the opposite side.
 
A typical tunneling scenario involves a finite potential barrier of Height V_0 and width L. When a particle of energy E < V_0 approaches the barrier, part of its wavefunction decays inside the barrier yet continues far enough to "leak through", creating a transmitted wave.
In classical physics, a particle with energy lower than a barrier height cannot cross that barrier. For example, if a ball does not have enough kinetic energy to roll over a hill, it rolls back down. Quantum particles behave differently because they are represented by wavefunctions rather than by perfectly localized objects.
 
A typical tunneling scenario involves a finite potential barrier with height <math>V_0</math> and width <math>L</math>. If a particle with energy <math>E</math> approaches the barrier and <math>E < V_0</math>, classical mechanics predicts total reflection. Quantum mechanics predicts that part of the particle's wavefunction decays inside the barrier. If the barrier is not too wide, a small portion of the wavefunction can survive to the far side and become a transmitted wave.
 
In simple terms, tunneling does not mean that the particle "borrows energy" in a literal way. The more precise explanation is that the particle's wavefunction does not instantly become zero inside the barrier. Instead, it decreases exponentially. If enough of the wavefunction remains after crossing the barrier region, there is a nonzero probability of detecting the particle on the other side.


===Classical Expectations===
===Classical Expectations===
Under classical mechanics, the criterion for overcoming a barrier is simple: the particle's energy must exceed the potential energy of the barrier. If E < V_0, the particle should reflect entirely. No motion is predicted within the barrier, and transmission probability is strictly zero.
 
Under classical mechanics, a particle can cross a potential barrier only if its total energy is greater than the barrier's potential energy. Therefore:
 
* If <math>E > V_0</math>, the particle can pass over the barrier.
* If <math>E < V_0</math>, the particle cannot pass through or over the barrier.
* The classical transmission probability for <math>E < V_0</math> is exactly zero.
 
This classical model works well for macroscopic objects, but it fails for microscopic particles such as electrons, protons, and alpha particles.


===Quantum Theory===
===Quantum Theory===
====Wave Behavior of Particles====
====Wave Behavior of Particles====
Quantum mechanics models particles with wavefunctions that satisfy the time-independent Schrodinger equation. These wavefunctions can be decomposed into incident, reflected, and transmitted components. Because waves generally do not vanish abruptly at boundaries, the wavefunction must extend into the barrier. Within this region, the wave amplitude decays exponentially.
 
When the decaying wave reaches the far side of the barrier, a small fraction survives and becomes a traveling wave in the output region. This survival mechanism produces tunneling.
Quantum mechanics describes particles using wavefunctions. A wavefunction, usually written as <math>\psi(x)</math>, contains information about the probability of finding a particle at different positions. The probability density is given by
 
<math>|\psi(x)|^2</math>.
 
When a quantum particle reaches a finite potential barrier, its wavefunction must satisfy boundary conditions at the edges of the barrier. Since waves generally do not suddenly disappear at a boundary, the wavefunction extends into the barrier. Inside the barrier, the wavefunction has an exponentially decaying form. If the barrier is thin enough, the wavefunction can still have a nonzero value on the far side.
 
This is the origin of tunneling.
 
====Important Intuition====
 
Tunneling probability depends strongly on three main quantities:
 
* '''Barrier width <math>L</math>:''' Wider barriers greatly reduce tunneling.
* '''Barrier height <math>V_0</math>:''' Taller barriers reduce tunneling when the particle energy is fixed.
* '''Particle mass <math>m</math>:''' Heavier particles tunnel much less easily than lighter particles.
 
This is why tunneling is common for electrons and other microscopic particles, but not noticeable for everyday objects.


====Heisenberg Uncertainty Principle====
====Heisenberg Uncertainty Principle====
A qualitative way to appreciate tunneling is through the energy-time variant of the Heisenberg uncertainty principle. Over extremely short time intervals, a particle's energy cannot be known exactly. This uncertainty permits temporary fluctuations that may allow the particle to explore regions classically inaccessible to it. While not a precise calculation tool, this principle offers intuitive support for why penetration into forbidden regions is possible.
 
The Heisenberg uncertainty principle is sometimes used as a qualitative way to think about tunneling, but it should not be treated as the main calculation method. Tunneling is not best understood as a particle simply "borrowing energy" and then paying it back. The more accurate explanation comes from solving the Schrödinger equation with the correct boundary conditions.
 
The uncertainty principle is still useful conceptually because it reminds us that quantum particles are not perfectly localized objects with perfectly definite classical trajectories.


====The Schrödinger Equation====
====The Schrödinger Equation====
The [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation] tells us that a wave function must be continuous at each boundary it encounters, and the derivative of the wave function must also be continuous except when the boundary height is infinite. Applying this to a finite potential barrier, we can find the probability a particle can tunnel through a potential barrier. Notably, the wave functions and calculations most resemble the [[Solution for a Single Particle in a Semi-Infinite Quantum Well | solution for a single particle in a semi-infinite well]] when <math>V_0 > E </math> in regions II and III. 


===Mathematical Model===
The time-independent Schrödinger equation determines the allowed wavefunctions for a particle in a given potential energy function:
Beyond the conceptual understanding of how quantum tunneling works, calculations can be made to determine the probability a wave function can tunnel through a specific finite potential barrier. This is known as transmission probability and to understand this equation, we must look at the wave equations associated with the potential barrier.
 
<math>-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x)</math>.
 
For a finite potential barrier, the wavefunction must be continuous at each boundary. Its derivative must also be continuous as long as the potential is finite. These conditions allow us to solve for the reflected and transmitted parts of the wave.
 
==Mathematical Model==
 
Consider a rectangular potential barrier:
 
<math>
V(x)=
\begin{cases}
0, & x<0 \\
V_0, & 0\le x\le L \\
0, & x>L
\end{cases}
</math>
 
Assume the particle energy satisfies
 
<math>0<E<V_0</math>.
 
This means the particle does not have enough energy to cross the barrier classically.
 
===Wave Equations===


====Wave Equations====
In each region, the Schrödinger equation has a different form.
The time independent Schrodinger equation is given by the following equation.


<math>-\frac{\hbar^2}{2m}\frac{d^2 \psi(x)}{dx^2} + V(x)\psi (x) = E \psi(x)</math>
'''Region I: Before the barrier, <math>x<0</math>'''


As mentioned before, the finite potential barrier creates three different regions with their own unique wave equations. Let's assume that the potential barrier is from <math>0 \le x \le L</math>. Given that, we can define the potential as follows:
<math>-\frac{\hbar^2}{2m}\frac{d^2\psi_I}{dx^2}=E\psi_I</math>


<math> V(x) = \begin{cases}
The general solution is
      0 & x < 0 \\
      V_0 & 0\leq x\leq L \\
      0 & x> L
  \end{cases} </math>


Thus, we can write the Schrodinger equations for the three regions.
<math>\psi_I(x)=Ae^{ikx}+Be^{-ikx}</math>.


* '''''Region I''''' : <math> -\frac{\hbar^2}{2m}\frac{d^2 \psi(x)}{dx^2} = E \psi(x) </math>
Here:


* '''''Region II''''' : <math> -\frac{\hbar^2}{2m}\frac{d^2 \psi(x)}{dx^2} + V_0 \psi (x) = E \psi(x) </math>
* <math>Ae^{ikx}</math> is the incident wave moving toward the barrier.
* <math>Be^{-ikx}</math> is the reflected wave moving away from the barrier.


* '''''Region III''''' : <math> -\frac{\hbar^2}{2m}\frac{d^2 \psi(x)}{dx^2} = E \psi(x) </math>
'''Region II: Inside the barrier, <math>0\le x\le L</math>'''


Boundary conditions require that the wave functions and their derivatives are continuous on each boundary i.e <math> x = 0 </math> and <math> x = L </math>. The general solution for each region,
<math>-\frac{\hbar^2}{2m}\frac{d^2\psi_{II}}{dx^2}+V_0\psi_{II}=E\psi_{II}</math>


* '''''Region I''''': <math> \psi(x)_I = Ae^{ikx} + Be^{-ikx} </math>
Since <math>E<V_0</math>, the solution is exponential rather than sinusoidal:


* '''''Region II''''': <math> \psi(x)_{II} = Ce^{-\alpha x} + De^{\alpha x } </math>
<math>\psi_{II}(x)=Ce^{-\kappa x}+De^{\kappa x}</math>.


* '''''Region III''''': <math> \psi(x)_{III} = Fe^{ikx} + Ge^{-ikx} </math>
This shows that the wavefunction decays inside the barrier instead of oscillating like a free traveling wave.


where <math> \alpha = \sqrt {\frac{2m}{\hbar^2} (V_0 - E)} </math> and <math> k = \sqrt {\frac{2m}{\hbar^2} E} </math>.
'''Region III: After the barrier, <math>x>L</math>'''


There is only one direction in which the wave in region III is traveling (<math> +\infty </math>). Thus the wave functions are now, 
<math>-\frac{\hbar^2}{2m}\frac{d^2\psi_{III}}{dx^2}=E\psi_{III}</math>


* '''''Region I''''': <math> \psi(x)_I = Ae^{ikx} + Be^{-ikx} </math>
Since there is no incoming wave from the far right, the transmitted wave is


* '''''Region II''''': <math> \psi(x)_{II} = Ce^{-\alpha x} + De^{\alpha x } </math>
<math>\psi_{III}(x)=Fe^{ikx}</math>.


* '''''Region III''''': <math> \psi(x)_{III} = Fe^{ikx} </math>
The constants are


It is important to note that the constants A, B, C, D, and F refer to the amplitude of the five waves in this problem:
<math>k=\frac{\sqrt{2mE}}{\hbar}</math>
* A: incident wave at <math> x = 0 </math>
* B: reflected wave at <math> x = 0 </math>
* C: incident wave at <math> x = L </math>
* D: reflected wave at <math> x = L </math>
* F: transmitted wave


====Tunneling Probability====
and
The probability that a particle can tunnel through a potential barrier utilizes the wave functions discussed in the previous section. Specifically, tunneling probability is the ratio of the transmitted wave intensity <math> |F|^2</math> and the incident wave intensity <math> |A|^2 </math>. To solve for these constants, we must apply boundary conditions.


At <math> x = 0 </math>, we know region I and II must agree and create a continuous wave function.
<math>\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}</math>.


<math> Ae^{ikx} + Be^{-ikx} = Ce^{-\alpha x } + De^{\alpha x} </math>
The quantity <math>k</math> describes the oscillating wave outside the barrier. The quantity <math>\kappa</math> describes the exponential decay inside the barrier.


<math> \frac{d}{dx}(Ae^{ikx} + Be^{-ikx} = Ce^{-\alpha x } + De^{\alpha x}) </math>
===Boundary Conditions===


<math> ik(Ae^{ikx} - Be^{-ikx}) = \alpha(-Ce^{-\alpha x } + De^{\alpha x}) </math>
The wavefunction and its derivative must match at both edges of the barrier.


The same applies at <math> x = L </math>.
At <math>x=0</math>:


<math> Ce^{-\alpha x } + De^{\alpha x} = Fe^{ikx} </math>
<math>\psi_I(0)=\psi_{II}(0)</math>


<math> \frac{d}{dx}(Ce^{-\alpha x } + De^{\alpha x} = Fe^{ikx}) </math>
<math>\frac{d\psi_I}{dx}\bigg|_{x=0}=\frac{d\psi_{II}}{dx}\bigg|_{x=0}</math>


<math> \alpha(-Ce^{-\alpha x } + De^{\alpha x}) = ikFe^{ikx} </math>
At <math>x=L</math>:


After some lengthy algebra and calculations, we know that the ratio of the transmitted wave amplitude and incident wave amplitude is this messy expression.
<math>\psi_{II}(L)=\psi_{III}(L)</math>
 
<math>\frac{d\psi_{II}}{dx}\bigg|_{x=L}=\frac{d\psi_{III}}{dx}\bigg|_{x=L}</math>
 
These four equations can be solved for the unknown amplitudes <math>B</math>, <math>C</math>, <math>D</math>, and <math>F</math> in terms of the incoming amplitude <math>A</math>.
 
===Tunneling Probability===
 
The tunneling probability is also called the transmission coefficient. It is written as
 
<math>T=\frac{\text{transmitted intensity}}{\text{incident intensity}}</math>.
 
For a rectangular barrier with <math>E<V_0</math>, the exact transmission coefficient is


<math>
<math>
\frac{F}{A} = \frac{e^{-ikL}}{cosh(\alpha L) +i(\gamma /2)sinh(\alpha L)} </math>[https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.07%3A_Quantum_Tunneling_of_Particles_through_Potential_Barriers]
T=\left[1+\frac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1}
</math>.
 
For a thick or high barrier, where <math>\kappa L</math> is large, a useful approximation is


where <math> \gamma = \frac{\alpha}{k} - \frac{k}{\alpha} </math>.
<math>
T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L}
</math>.


We know that tunneling probability is a ratio of intensity. So when we multiply the ratio of amplitudes by the conjugate, we get the following equation for tunneling probability.
This approximation shows the most important idea:


<math> T \approx 16 \frac{E}{V_0} (1-\frac{E}{V_0}) e^{-2kL} </math>
<math>T</math> decreases exponentially as the barrier becomes wider or harder to penetrate.


where,
Since the exponential term is <math>e^{-2\kappa L}</math>, even a small increase in barrier width can cause a huge decrease in tunneling probability.
<math> k = \sqrt {\frac{2m(V_0-E)}{\hbar^2}} </math> and L is the width of the barrier.


===Computational Model===
==Computational Model==
The following simulation illustrates a wave function for various potentials in both plane wave and wave packet form.  
 
The PhET quantum tunneling simulation lets users adjust the energy of a particle and the shape of the potential barrier. This is useful because tunneling is easier to understand when the wavefunction can be seen visually.


<iframe src="https://phet.colorado.edu/sims/cheerpj/quantum-tunneling/latest/quantum-tunneling.html?simulation=quantum-tunneling"
<iframe src="https://phet.colorado.edu/sims/cheerpj/quantum-tunneling/latest/quantum-tunneling.html?simulation=quantum-tunneling"
Line 110: Line 173:
         height="600"
         height="600"
         allowfullscreen>
         allowfullscreen>
</iframe> [https://phet.colorado.edu/en/simulations/quantum-tunneling]
</iframe>
 
When using the simulation, try the following:
 
* Set the particle energy below the barrier height.
* Increase the barrier width and observe how the transmitted wave decreases.
* Increase the barrier height and observe how the transmitted wave decreases.
* Compare a plane wave with a wave packet.
 
==Common Exam Mistakes==
 
* '''Using classical reasoning only.''' In classical physics, <math>E<V_0</math> means zero transmission. In quantum mechanics, the transmission probability can be small but nonzero.
 
* '''Forgetting the exponential dependence.''' Tunneling probability is extremely sensitive to barrier width because of the factor <math>e^{-2\kappa L}</math>.
 
* '''Confusing <math>k</math> and <math>\kappa</math>.''' The quantity <math>k</math> describes oscillation outside the barrier. The quantity <math>\kappa</math> describes exponential decay inside the barrier.
 
* '''Thinking tunneling means the particle literally climbs over the barrier.''' The particle is not following a classical path over the barrier. The wavefunction extends through the barrier.
 
* '''Ignoring mass.''' Heavier particles have larger <math>\kappa</math>, so they tunnel much less easily.


==Examples==
==Examples==
===Simple===
===Simple===


A particle has 5.0 eV of energy. What is the probability that it will tunnel through a potential barrier of 10.0 eV with a width/thickness of 1.00 nm?
An electron has energy <math>5.0\text{ eV}</math>. What is the approximate probability that it will tunnel through a potential barrier of height <math>10.0\text{ eV}</math> and width <math>1.00\text{ nm}</math>?


'''''Solution'''''
'''''Solution'''''


First, let's define the given variables and equation we must use
Given:
* <math> E = 5.0 eV </math>
* <math> V_0 = 10.0 eV</math>
* <math> L = 1.00 nm = 1.00 \times 10^{-9} m </math>


*<math> T \approx 16\frac {E}{V_0} (1-\frac{E}{V_0})e^{-2kL} </math>
* <math>E=5.0\text{ eV}</math>
* <math> k = \sqrt{\frac {2m(V_0 -E)}{\hbar^2}} </math>
* <math>V_0=10.0\text{ eV}</math>
* <math>L=1.00\text{ nm}=1.00\times10^{-9}\text{ m}</math>
* <math>m_e=9.11\times10^{-31}\text{ kg}</math>
* <math>\hbar=1.055\times10^{-34}\text{ J}\cdot\text{s}</math>
* <math>1\text{ eV}=1.602\times10^{-19}\text{ J}</math>


Next, plug in our given values into the equation and solve.
First calculate <math>\kappa</math>:


<math> k = \sqrt {\frac {2(9.11 \times 10^{-31} kg)(10.0-5.0 eV)(1.602 \times 10^{-19} J)}{(1.055 \times 10^{-34})^2}} = 1.145 \times 10^{10} m^{-1} </math>
<math>
\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}
</math>


<math> T \approx 16\frac {5.0}{10.0} (1-\frac{5.0}{10.0})e^{-2(1.145 \times 10^{10} m^{-1})(1.00 \times 10^{-9})} = 4.52 \times 10^{-10} </math>
<math>
\kappa=
\frac{\sqrt{2(9.11\times10^{-31})(5.0)(1.602\times10^{-19})}}{1.055\times10^{-34}}
</math>
 
<math>
\kappa\approx1.15\times10^{10}\text{ m}^{-1}
</math>
 
Now use the approximation:
 
<math>
T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L}
</math>
 
<math>
T\approx 16\left(\frac{5.0}{10.0}\right)\left(1-\frac{5.0}{10.0}\right)e^{-2(1.15\times10^{10})(1.00\times10^{-9})}
</math>
 
<math>
T\approx4.5\times10^{-10}
</math>
 
So the tunneling probability is approximately
 
<math>
\boxed{4.5\times10^{-10}}
</math>.
 
This is a very small probability, but it is not zero.


===Middling===
===Middling===
An electron with energy <math>4.0\text{ eV}</math> approaches a rectangular barrier of height <math>8.0\text{ eV}</math>. Compare the tunneling probability when the barrier width is <math>0.50\text{ nm}</math> and when the barrier width is <math>1.00\text{ nm}</math>.
'''''Solution'''''
Given:
* <math>E=4.0\text{ eV}</math>
* <math>V_0=8.0\text{ eV}</math>
* <math>V_0-E=4.0\text{ eV}</math>
Calculate <math>\kappa</math>:
<math>
\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}
</math>
<math>
\kappa\approx1.02\times10^{10}\text{ m}^{-1}
</math>
The approximate tunneling probability is
<math>
T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L}
</math>.
Since <math>E/V_0=4.0/8.0=0.50</math>,
<math>
16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)=16(0.50)(0.50)=4
</math>.
For <math>L=0.50\text{ nm}</math>:
<math>
T\approx 4e^{-2(1.02\times10^{10})(0.50\times10^{-9})}
</math>
<math>
T\approx1.4\times10^{-4}
</math>.
For <math>L=1.00\text{ nm}</math>:
<math>
T\approx 4e^{-2(1.02\times10^{10})(1.00\times10^{-9})}
</math>
<math>
T\approx5.0\times10^{-9}
</math>.
Doubling the width does not merely cut the probability in half. It reduces the tunneling probability by many orders of magnitude because the width appears inside an exponential.
===Difficult===
===Difficult===
An electron with energy <math>2.0\text{ eV}</math> approaches a rectangular barrier of height <math>5.0\text{ eV}</math>. What barrier width gives an approximate tunneling probability of <math>1.0\times10^{-6}</math>?
'''''Solution'''''
Given:
* <math>E=2.0\text{ eV}</math>
* <math>V_0=5.0\text{ eV}</math>
* <math>T=1.0\times10^{-6}</math>
* <math>V_0-E=3.0\text{ eV}</math>
Start with the approximate tunneling equation:
<math>
T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L}
</math>.
Calculate the coefficient:
<math>
16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)
=
16\left(\frac{2.0}{5.0}\right)\left(1-\frac{2.0}{5.0}\right)
</math>
<math>
=16(0.40)(0.60)=3.84
</math>.
So
<math>
1.0\times10^{-6}=3.84e^{-2\kappa L}
</math>.
Divide both sides by <math>3.84</math>:
<math>
e^{-2\kappa L}=2.60\times10^{-7}
</math>.
Take the natural logarithm of both sides:
<math>
-2\kappa L=\ln(2.60\times10^{-7})
</math>
<math>
-2\kappa L\approx -15.16
</math>.
Now solve for <math>L</math>:
<math>
L=\frac{15.16}{2\kappa}
</math>.
Calculate <math>\kappa</math>:
<math>
\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}
</math>
<math>
\kappa=
\frac{\sqrt{2(9.11\times10^{-31})(3.0)(1.602\times10^{-19})}}{1.055\times10^{-34}}
</math>
<math>
\kappa\approx8.87\times10^9\text{ m}^{-1}
</math>.
Therefore,
<math>
L=\frac{15.16}{2(8.87\times10^9)}
</math>
<math>
L\approx8.55\times10^{-10}\text{ m}
</math>.
So the barrier width is approximately
<math>
\boxed{0.855\text{ nm}}
</math>.


==Applications==
==Applications==
'''''Nuclear Fusion in Stars'''''


Quantum tunneling enables protons to overcome their mutual electrostatic repulsion, allowing fusion reactions to occur at stellar core temperatures.
====Nuclear Fusion in Stars====
 
Quantum tunneling helps explain how nuclear fusion can occur inside stars. Protons repel each other because they are positively charged. Classically, many protons in a star's core would not have enough energy to overcome this electric repulsion. Quantum tunneling gives them a small probability of getting close enough for the strong nuclear force to bind them together.


'''''Scanning Tunneling Microscopy (STM)'''''
====Alpha Decay====


Tunneling current between a sharp metallic tip and a surface allows imaging of materials with atomic resolution.
Alpha decay occurs when an unstable nucleus emits an alpha particle. The alpha particle is trapped inside the nucleus by a potential barrier. Classically, it may not have enough energy to escape. Quantum mechanically, it can tunnel through the barrier and leave the nucleus.


'''''Semiconductor Devices'''''
====Scanning Tunneling Microscopy====


Tunnel diodes and modern transistors exploit controlled tunneling currents.
A scanning tunneling microscope, or STM, uses tunneling current to image surfaces at the atomic scale. When a sharp conducting tip is brought very close to a surface, electrons can tunnel between the tip and the surface. The tunneling current is extremely sensitive to distance, which allows the STM to detect tiny changes in surface height.


==Historical Background==
====Semiconductor Devices====
The notion of barrier penetration was introduced in the late 1920s by physicist Friedrich Hund, who noticed that quantum wavefunctions could leak between adjacent molecular wells. Soon after, George Gamow applied tunneling to explain alpha decay, firmly establishing it as a central quantum phenomenon.
 
Some semiconductor devices rely on tunneling. Tunnel diodes, flash memory, and very small transistors can involve tunneling effects. As electronic devices become smaller, tunneling becomes more important because electrons interact with barriers only a few nanometers wide.
 
==Connection to Matter and Interactions==
 
In Matter and Interactions, many problems focus on the relationship between energy, force, and motion. Quantum tunneling is a case where the classical energy model is not enough by itself. Classically, a particle with <math>E<V_0</math> cannot pass the barrier. Quantum mechanically, the wavefunction still exists inside the barrier and may continue to the far side.


==See Also==
This makes tunneling a useful example of how quantum mechanics changes the meaning of motion. Instead of asking only whether a particle has enough energy to cross a barrier, we ask what probability the wavefunction gives for finding the particle beyond the barrier.
[[Heisenberg Uncertainty Principle]]


[[Wave-Particle Duality]]
==Historical Background==


[[Solution for a Single Particle in a Semi-Infinite Quantum Well]]
Quantum tunneling was developed during the early years of quantum mechanics. Friedrich Hund discussed wavefunction penetration in molecular systems in the 1920s. George Gamow later applied tunneling to alpha decay, showing that quantum mechanics could explain how alpha particles escape from atomic nuclei. Tunneling soon became one of the clearest examples of how quantum mechanics differs from classical mechanics.


[[Application of Statistics in Physics]]
==See Also==


===Related Topics===
* [[Heisenberg Uncertainty Principle]]
* [[Heisenberg Uncertainty Principle]]
* [[ Wave-Particle Duality]]
* [[Wave-Particle Duality]]
* [[ Schrodinger Equation]]
* [[Schrodinger Equation]]
* [[Quantum Wells and Bound States]]
* [[Quantum Wells and Bound States]]
* [[Solution for a Single Particle in a Semi-Infinite Quantum Well]]
* [[Application of Statistics in Physics]]
==External Resources==


===External Resources===
* [https://phet.colorado.edu/en/simulations/quantum-tunneling PhET: Quantum Tunneling and Wave Packets]
* [https://phet.colorado.edu/en/simulations/quantum-tunneling]
* [https://openstax.org/details/books/university-physics-volume-3 OpenStax University Physics Volume 3]
* [https://www.youtube.com/watch?v=cTodS8hkSDg]
* [https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/University_Physics_III_-_Optics_and_Modern_Physics_%28OpenStax%29/07%3A_Quantum_Mechanics/7.07%3A_Quantum_Tunneling_of_Particles_through_Potential_Barriers LibreTexts: Quantum Tunneling of Particles through Potential Barriers]
* [https://www.youtube.com/watch?v=RF7dDt3tVmI]
* [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html HyperPhysics: Barrier Penetration]


==References==
==References==
* https://openstax.org/details/books/university-physics-volume-3
 
* http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
* OpenStax, ''University Physics Volume 3'', Chapter 7: Quantum Mechanics.
* https://physicstoday.scitation.org/doi/10.1063/1.1510281
* LibreTexts, ''Quantum Tunneling of Particles through Potential Barriers''.
* https://phet.colorado.edu/en/simulations/quantum-tunneling
* HyperPhysics, ''Barrier Penetration''.
* PhET Interactive Simulations, ''Quantum Tunneling and Wave Packets''.


[[Category: Quantum Mechanics]]
[[Category: Quantum Mechanics]]

Revision as of 12:25, 27 April 2026

Page Claimed By: Haripratiik Arunkumar Malarkodi Spring '26

Quantum tunneling is a fundamental effect in quantum mechanics in which a particle has a nonzero probability of passing through a potential barrier even when its classical energy is too small to go over the barrier. In classical mechanics, this would be impossible. In quantum mechanics, particles are described by wavefunctions, and those wavefunctions can extend into regions that would be classically forbidden.

Quantum tunneling is important because it explains physical processes that classical physics cannot explain, including alpha decay, nuclear fusion in stars, scanning tunneling microscopy, and tunneling effects in semiconductor devices.

Overview

In classical physics, a particle with energy lower than a barrier height cannot cross that barrier. For example, if a ball does not have enough kinetic energy to roll over a hill, it rolls back down. Quantum particles behave differently because they are represented by wavefunctions rather than by perfectly localized objects.

A typical tunneling scenario involves a finite potential barrier with height [math]\displaystyle{ V_0 }[/math] and width [math]\displaystyle{ L }[/math]. If a particle with energy [math]\displaystyle{ E }[/math] approaches the barrier and [math]\displaystyle{ E \lt V_0 }[/math], classical mechanics predicts total reflection. Quantum mechanics predicts that part of the particle's wavefunction decays inside the barrier. If the barrier is not too wide, a small portion of the wavefunction can survive to the far side and become a transmitted wave.

In simple terms, tunneling does not mean that the particle "borrows energy" in a literal way. The more precise explanation is that the particle's wavefunction does not instantly become zero inside the barrier. Instead, it decreases exponentially. If enough of the wavefunction remains after crossing the barrier region, there is a nonzero probability of detecting the particle on the other side.

Classical Expectations

Under classical mechanics, a particle can cross a potential barrier only if its total energy is greater than the barrier's potential energy. Therefore:

  • If [math]\displaystyle{ E \gt V_0 }[/math], the particle can pass over the barrier.
  • If [math]\displaystyle{ E \lt V_0 }[/math], the particle cannot pass through or over the barrier.
  • The classical transmission probability for [math]\displaystyle{ E \lt V_0 }[/math] is exactly zero.

This classical model works well for macroscopic objects, but it fails for microscopic particles such as electrons, protons, and alpha particles.

Quantum Theory

Wave Behavior of Particles

Quantum mechanics describes particles using wavefunctions. A wavefunction, usually written as [math]\displaystyle{ \psi(x) }[/math], contains information about the probability of finding a particle at different positions. The probability density is given by

[math]\displaystyle{ |\psi(x)|^2 }[/math].

When a quantum particle reaches a finite potential barrier, its wavefunction must satisfy boundary conditions at the edges of the barrier. Since waves generally do not suddenly disappear at a boundary, the wavefunction extends into the barrier. Inside the barrier, the wavefunction has an exponentially decaying form. If the barrier is thin enough, the wavefunction can still have a nonzero value on the far side.

This is the origin of tunneling.

Important Intuition

Tunneling probability depends strongly on three main quantities:

  • Barrier width [math]\displaystyle{ L }[/math]: Wider barriers greatly reduce tunneling.
  • Barrier height [math]\displaystyle{ V_0 }[/math]: Taller barriers reduce tunneling when the particle energy is fixed.
  • Particle mass [math]\displaystyle{ m }[/math]: Heavier particles tunnel much less easily than lighter particles.

This is why tunneling is common for electrons and other microscopic particles, but not noticeable for everyday objects.

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle is sometimes used as a qualitative way to think about tunneling, but it should not be treated as the main calculation method. Tunneling is not best understood as a particle simply "borrowing energy" and then paying it back. The more accurate explanation comes from solving the Schrödinger equation with the correct boundary conditions.

The uncertainty principle is still useful conceptually because it reminds us that quantum particles are not perfectly localized objects with perfectly definite classical trajectories.

The Schrödinger Equation

The time-independent Schrödinger equation determines the allowed wavefunctions for a particle in a given potential energy function:

[math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x) }[/math].

For a finite potential barrier, the wavefunction must be continuous at each boundary. Its derivative must also be continuous as long as the potential is finite. These conditions allow us to solve for the reflected and transmitted parts of the wave.

Mathematical Model

Consider a rectangular potential barrier:

[math]\displaystyle{ V(x)= \begin{cases} 0, & x\lt 0 \\ V_0, & 0\le x\le L \\ 0, & x\gt L \end{cases} }[/math]

Assume the particle energy satisfies

[math]\displaystyle{ 0\lt E\lt V_0 }[/math].

This means the particle does not have enough energy to cross the barrier classically.

Wave Equations

In each region, the Schrödinger equation has a different form.

Region I: Before the barrier, [math]\displaystyle{ x\lt 0 }[/math]

[math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2\psi_I}{dx^2}=E\psi_I }[/math]

The general solution is

[math]\displaystyle{ \psi_I(x)=Ae^{ikx}+Be^{-ikx} }[/math].

Here:

  • [math]\displaystyle{ Ae^{ikx} }[/math] is the incident wave moving toward the barrier.
  • [math]\displaystyle{ Be^{-ikx} }[/math] is the reflected wave moving away from the barrier.

Region II: Inside the barrier, [math]\displaystyle{ 0\le x\le L }[/math]

[math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2\psi_{II}}{dx^2}+V_0\psi_{II}=E\psi_{II} }[/math]

Since [math]\displaystyle{ E\lt V_0 }[/math], the solution is exponential rather than sinusoidal:

[math]\displaystyle{ \psi_{II}(x)=Ce^{-\kappa x}+De^{\kappa x} }[/math].

This shows that the wavefunction decays inside the barrier instead of oscillating like a free traveling wave.

Region III: After the barrier, [math]\displaystyle{ x\gt L }[/math]

[math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2\psi_{III}}{dx^2}=E\psi_{III} }[/math]

Since there is no incoming wave from the far right, the transmitted wave is

[math]\displaystyle{ \psi_{III}(x)=Fe^{ikx} }[/math].

The constants are

[math]\displaystyle{ k=\frac{\sqrt{2mE}}{\hbar} }[/math]

and

[math]\displaystyle{ \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar} }[/math].

The quantity [math]\displaystyle{ k }[/math] describes the oscillating wave outside the barrier. The quantity [math]\displaystyle{ \kappa }[/math] describes the exponential decay inside the barrier.

Boundary Conditions

The wavefunction and its derivative must match at both edges of the barrier.

At [math]\displaystyle{ x=0 }[/math]:

[math]\displaystyle{ \psi_I(0)=\psi_{II}(0) }[/math]

[math]\displaystyle{ \frac{d\psi_I}{dx}\bigg|_{x=0}=\frac{d\psi_{II}}{dx}\bigg|_{x=0} }[/math]

At [math]\displaystyle{ x=L }[/math]:

[math]\displaystyle{ \psi_{II}(L)=\psi_{III}(L) }[/math]

[math]\displaystyle{ \frac{d\psi_{II}}{dx}\bigg|_{x=L}=\frac{d\psi_{III}}{dx}\bigg|_{x=L} }[/math]

These four equations can be solved for the unknown amplitudes [math]\displaystyle{ B }[/math], [math]\displaystyle{ C }[/math], [math]\displaystyle{ D }[/math], and [math]\displaystyle{ F }[/math] in terms of the incoming amplitude [math]\displaystyle{ A }[/math].

Tunneling Probability

The tunneling probability is also called the transmission coefficient. It is written as

[math]\displaystyle{ T=\frac{\text{transmitted intensity}}{\text{incident intensity}} }[/math].

For a rectangular barrier with [math]\displaystyle{ E\lt V_0 }[/math], the exact transmission coefficient is

[math]\displaystyle{ T=\left[1+\frac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1} }[/math].

For a thick or high barrier, where [math]\displaystyle{ \kappa L }[/math] is large, a useful approximation is

[math]\displaystyle{ T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L} }[/math].

This approximation shows the most important idea:

[math]\displaystyle{ T }[/math] decreases exponentially as the barrier becomes wider or harder to penetrate.

Since the exponential term is [math]\displaystyle{ e^{-2\kappa L} }[/math], even a small increase in barrier width can cause a huge decrease in tunneling probability.

Computational Model

The PhET quantum tunneling simulation lets users adjust the energy of a particle and the shape of the potential barrier. This is useful because tunneling is easier to understand when the wavefunction can be seen visually.

<iframe src="https://phet.colorado.edu/sims/cheerpj/quantum-tunneling/latest/quantum-tunneling.html?simulation=quantum-tunneling" 

       width="800"
       height="600"
       allowfullscreen>

</iframe>

When using the simulation, try the following:

  • Set the particle energy below the barrier height.
  • Increase the barrier width and observe how the transmitted wave decreases.
  • Increase the barrier height and observe how the transmitted wave decreases.
  • Compare a plane wave with a wave packet.

Common Exam Mistakes

  • Using classical reasoning only. In classical physics, [math]\displaystyle{ E\lt V_0 }[/math] means zero transmission. In quantum mechanics, the transmission probability can be small but nonzero.
  • Forgetting the exponential dependence. Tunneling probability is extremely sensitive to barrier width because of the factor [math]\displaystyle{ e^{-2\kappa L} }[/math].
  • Confusing [math]\displaystyle{ k }[/math] and [math]\displaystyle{ \kappa }[/math]. The quantity [math]\displaystyle{ k }[/math] describes oscillation outside the barrier. The quantity [math]\displaystyle{ \kappa }[/math] describes exponential decay inside the barrier.
  • Thinking tunneling means the particle literally climbs over the barrier. The particle is not following a classical path over the barrier. The wavefunction extends through the barrier.
  • Ignoring mass. Heavier particles have larger [math]\displaystyle{ \kappa }[/math], so they tunnel much less easily.

Examples

Simple

An electron has energy [math]\displaystyle{ 5.0\text{ eV} }[/math]. What is the approximate probability that it will tunnel through a potential barrier of height [math]\displaystyle{ 10.0\text{ eV} }[/math] and width [math]\displaystyle{ 1.00\text{ nm} }[/math]?

Solution

Given:

  • [math]\displaystyle{ E=5.0\text{ eV} }[/math]
  • [math]\displaystyle{ V_0=10.0\text{ eV} }[/math]
  • [math]\displaystyle{ L=1.00\text{ nm}=1.00\times10^{-9}\text{ m} }[/math]
  • [math]\displaystyle{ m_e=9.11\times10^{-31}\text{ kg} }[/math]
  • [math]\displaystyle{ \hbar=1.055\times10^{-34}\text{ J}\cdot\text{s} }[/math]
  • [math]\displaystyle{ 1\text{ eV}=1.602\times10^{-19}\text{ J} }[/math]

First calculate [math]\displaystyle{ \kappa }[/math]:

[math]\displaystyle{ \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar} }[/math]

[math]\displaystyle{ \kappa= \frac{\sqrt{2(9.11\times10^{-31})(5.0)(1.602\times10^{-19})}}{1.055\times10^{-34}} }[/math]

[math]\displaystyle{ \kappa\approx1.15\times10^{10}\text{ m}^{-1} }[/math]

Now use the approximation:

[math]\displaystyle{ T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L} }[/math]

[math]\displaystyle{ T\approx 16\left(\frac{5.0}{10.0}\right)\left(1-\frac{5.0}{10.0}\right)e^{-2(1.15\times10^{10})(1.00\times10^{-9})} }[/math]

[math]\displaystyle{ T\approx4.5\times10^{-10} }[/math]

So the tunneling probability is approximately

[math]\displaystyle{ \boxed{4.5\times10^{-10}} }[/math].

This is a very small probability, but it is not zero.

Middling

An electron with energy [math]\displaystyle{ 4.0\text{ eV} }[/math] approaches a rectangular barrier of height [math]\displaystyle{ 8.0\text{ eV} }[/math]. Compare the tunneling probability when the barrier width is [math]\displaystyle{ 0.50\text{ nm} }[/math] and when the barrier width is [math]\displaystyle{ 1.00\text{ nm} }[/math].

Solution

Given:

  • [math]\displaystyle{ E=4.0\text{ eV} }[/math]
  • [math]\displaystyle{ V_0=8.0\text{ eV} }[/math]
  • [math]\displaystyle{ V_0-E=4.0\text{ eV} }[/math]

Calculate [math]\displaystyle{ \kappa }[/math]:

[math]\displaystyle{ \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar} }[/math]

[math]\displaystyle{ \kappa\approx1.02\times10^{10}\text{ m}^{-1} }[/math]

The approximate tunneling probability is

[math]\displaystyle{ T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L} }[/math].

Since [math]\displaystyle{ E/V_0=4.0/8.0=0.50 }[/math],

[math]\displaystyle{ 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)=16(0.50)(0.50)=4 }[/math].

For [math]\displaystyle{ L=0.50\text{ nm} }[/math]:

[math]\displaystyle{ T\approx 4e^{-2(1.02\times10^{10})(0.50\times10^{-9})} }[/math]

[math]\displaystyle{ T\approx1.4\times10^{-4} }[/math].

For [math]\displaystyle{ L=1.00\text{ nm} }[/math]:

[math]\displaystyle{ T\approx 4e^{-2(1.02\times10^{10})(1.00\times10^{-9})} }[/math]

[math]\displaystyle{ T\approx5.0\times10^{-9} }[/math].

Doubling the width does not merely cut the probability in half. It reduces the tunneling probability by many orders of magnitude because the width appears inside an exponential.

Difficult

An electron with energy [math]\displaystyle{ 2.0\text{ eV} }[/math] approaches a rectangular barrier of height [math]\displaystyle{ 5.0\text{ eV} }[/math]. What barrier width gives an approximate tunneling probability of [math]\displaystyle{ 1.0\times10^{-6} }[/math]?

Solution

Given:

  • [math]\displaystyle{ E=2.0\text{ eV} }[/math]
  • [math]\displaystyle{ V_0=5.0\text{ eV} }[/math]
  • [math]\displaystyle{ T=1.0\times10^{-6} }[/math]
  • [math]\displaystyle{ V_0-E=3.0\text{ eV} }[/math]

Start with the approximate tunneling equation:

[math]\displaystyle{ T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L} }[/math].

Calculate the coefficient:

[math]\displaystyle{ 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right) = 16\left(\frac{2.0}{5.0}\right)\left(1-\frac{2.0}{5.0}\right) }[/math]

[math]\displaystyle{ =16(0.40)(0.60)=3.84 }[/math].

So

[math]\displaystyle{ 1.0\times10^{-6}=3.84e^{-2\kappa L} }[/math].

Divide both sides by [math]\displaystyle{ 3.84 }[/math]:

[math]\displaystyle{ e^{-2\kappa L}=2.60\times10^{-7} }[/math].

Take the natural logarithm of both sides:

[math]\displaystyle{ -2\kappa L=\ln(2.60\times10^{-7}) }[/math]

[math]\displaystyle{ -2\kappa L\approx -15.16 }[/math].

Now solve for [math]\displaystyle{ L }[/math]:

[math]\displaystyle{ L=\frac{15.16}{2\kappa} }[/math].

Calculate [math]\displaystyle{ \kappa }[/math]:

[math]\displaystyle{ \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar} }[/math]

[math]\displaystyle{ \kappa= \frac{\sqrt{2(9.11\times10^{-31})(3.0)(1.602\times10^{-19})}}{1.055\times10^{-34}} }[/math]

[math]\displaystyle{ \kappa\approx8.87\times10^9\text{ m}^{-1} }[/math].

Therefore,

[math]\displaystyle{ L=\frac{15.16}{2(8.87\times10^9)} }[/math]

[math]\displaystyle{ L\approx8.55\times10^{-10}\text{ m} }[/math].

So the barrier width is approximately

[math]\displaystyle{ \boxed{0.855\text{ nm}} }[/math].

Applications

Nuclear Fusion in Stars

Quantum tunneling helps explain how nuclear fusion can occur inside stars. Protons repel each other because they are positively charged. Classically, many protons in a star's core would not have enough energy to overcome this electric repulsion. Quantum tunneling gives them a small probability of getting close enough for the strong nuclear force to bind them together.

Alpha Decay

Alpha decay occurs when an unstable nucleus emits an alpha particle. The alpha particle is trapped inside the nucleus by a potential barrier. Classically, it may not have enough energy to escape. Quantum mechanically, it can tunnel through the barrier and leave the nucleus.

Scanning Tunneling Microscopy

A scanning tunneling microscope, or STM, uses tunneling current to image surfaces at the atomic scale. When a sharp conducting tip is brought very close to a surface, electrons can tunnel between the tip and the surface. The tunneling current is extremely sensitive to distance, which allows the STM to detect tiny changes in surface height.

Semiconductor Devices

Some semiconductor devices rely on tunneling. Tunnel diodes, flash memory, and very small transistors can involve tunneling effects. As electronic devices become smaller, tunneling becomes more important because electrons interact with barriers only a few nanometers wide.

Connection to Matter and Interactions

In Matter and Interactions, many problems focus on the relationship between energy, force, and motion. Quantum tunneling is a case where the classical energy model is not enough by itself. Classically, a particle with [math]\displaystyle{ E\lt V_0 }[/math] cannot pass the barrier. Quantum mechanically, the wavefunction still exists inside the barrier and may continue to the far side.

This makes tunneling a useful example of how quantum mechanics changes the meaning of motion. Instead of asking only whether a particle has enough energy to cross a barrier, we ask what probability the wavefunction gives for finding the particle beyond the barrier.

Historical Background

Quantum tunneling was developed during the early years of quantum mechanics. Friedrich Hund discussed wavefunction penetration in molecular systems in the 1920s. George Gamow later applied tunneling to alpha decay, showing that quantum mechanics could explain how alpha particles escape from atomic nuclei. Tunneling soon became one of the clearest examples of how quantum mechanics differs from classical mechanics.

See Also

External Resources

References

  • OpenStax, University Physics Volume 3, Chapter 7: Quantum Mechanics.
  • LibreTexts, Quantum Tunneling of Particles through Potential Barriers.
  • HyperPhysics, Barrier Penetration.
  • PhET Interactive Simulations, Quantum Tunneling and Wave Packets.
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