# Rectangular potential barrier

The rectangular potential barrier (also called the square barrier) is a standard one-dimensional model in quantum mechanics in which the potential energy is a constant value \( V_0 \) over a finite interval and zero elsewhere, typically written as \( V(x)=0 \) for \( x<0 \), \( V_0 \) for \( 0 \le x \le L \), and \( 0 \) for \( x>L \)<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup>. It is the simplest exactly solvable model of quantum tunneling and wave-mechanical reflection, and it serves as the textbook prototype for the transmission/reflection problem<sup>[2](https://people.umass.edu/bvs/614_1D.pdf)</sup><sup> • </sup><sup>[3](https://bhattercollege.ac.in/old/E_Learning/Physics/Rectangular_potential_barrier_Sem_IV%28H%29_CC9T_05_04_2020.pdf)</sup>.

| Key fact | Detail |
|---|---|
| Definition | \( V(x)=V_0 \) on a finite interval of width \( L \), zero outside; solved by matching plane-wave and exponential solutions at the two edges<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup> |
| Transmission, \( E<V_0 \) | \( T=\left[1+\dfrac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1} \), with \( \kappa=\sqrt{2m(V_0-E)}/\hbar \)<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup> |
| Thick-barrier limit | \( T \approx 16\dfrac{E}{V_0}\left(1-\dfrac{E}{V_0}\right)e^{-2\kappa L} \); transmission falls exponentially with barrier width<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup> |
| Transmission, \( E>V_0 \) | \( T=1/\left[1+\dfrac{\sin^2(k'L)}{4(E/V)(E/V-1)}\right] \); the barrier is fully transparent when \( \sin(k'L)=0 \), i.e. \( k'=n\pi/L \)<sup>[4](https://web.stanford.edu/~oas/SI/QM/Atkins05.pdf)</sup> |
| Equal-energy case | At \( E=V_0 \), \( T=1/(1+\xi^2/4) \) in dimensionless form, so a wide barrier still reflects strongly<sup>[2](https://people.umass.edu/bvs/614_1D.pdf)</sup> |
| Applications | Field emission, the scanning tunneling microscope (invented 1981), and alpha decay modeled as a sequence of rectangular barriers<sup>[5](https://openstax.org/books/university-physics-volume-3/pages/7-6-the-quantum-tunneling-of-particles-through-potential-barriers)</sup><sup> • </sup><sup>[6](https://facultystaff.richmond.edu/~ggilfoyl/research/gilfoyle/node2.html)</sup> |

## Solving the model

The time-independent [Schrödinger equation](https://www.edgepedia.org/schrodinger-equation) is solved separately in the three regions, and the wavefunction and its first derivative are matched at the two edges of the barrier; continuity of the derivative holds because the potential is finite<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup>.

**Three regions.** For a particle of energy \( 0<E<V_0 \), the region before the barrier carries an incident wave \( Ae^{ikx} \) and a reflected wave \( Be^{-ikx} \), with \( k=\sqrt{2mE}/\hbar \). Inside the barrier the solution is exponential rather than sinusoidal, \( \psi_{II}(x)=Ce^{-\kappa x}+De^{\kappa x} \), with \( \kappa=\sqrt{2m(V_0-E)}/\hbar \), showing that the wavefunction decays inside the barrier instead of oscillating. Beyond the barrier only the transmitted wave \( Fe^{ikx} \) remains<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup>.

The four boundary-condition equations at \( x=0 \) and \( x=L \) determine the reflected and transmitted amplitudes in terms of the incident amplitude \( A \)<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup>. The transmission coefficient, also called the tunneling probability, is the ratio of transmitted to incident intensity, \( T=|F|^2/|A|^2 \)<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup>.

## Transmission below the barrier

For \( E<V_0 \) the exact result is

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

Because the added term is real and non-negative, \( T \) is strictly less than 1 but never zero, so a portion of the wave is always transmitted even though classical mechanics predicts complete reflection<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup><sup> • </sup><sup>[7](https://faculty.washington.edu/seattle/physics227/reading/ch4-virtual-book.pdf)</sup>.

**Exponential sensitivity.** For a thick or high barrier, where \( \kappa L \) is large, the expression reduces to \( T \approx 16\dfrac{E}{V_0}\left(1-\dfrac{E}{V_0}\right)e^{-2\kappa L} \). The controlling factor is \( e^{-2\kappa L} \), so even a small increase in barrier width causes a large decrease in tunneling probability<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup>. A worked example with an electron at \( E=4.0 \) eV against a barrier of \( V_0=8.0 \) eV gives \( \kappa \approx 1.02 \times 10^{10} \ \mathrm{m}^{-1} \), so a nanometer-scale width already suppresses \( T \) to very small values<sup>[1](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)</sup>.

The same dependence on the product of width and height appears in the dimensionless parameter \( \xi = a\sqrt{2mU_0}/\hbar \), which combines the barrier's length and strength into one number; a long enough barrier becomes barely penetrable whenever \( E<U_0 \)<sup>[2](https://people.umass.edu/bvs/614_1D.pdf)</sup>.

## Transmission above the barrier

Classically, a particle with \( E>V_0 \) passes over the barrier freely. Quantum mechanically, reflection still occurs. Replacing \( \kappa \) by \( -ik' \) in the sub-barrier result gives<sup>[4](https://web.stanford.edu/~oas/SI/QM/Atkins05.pdf)</sup>

\[ T=\frac{1}{1+\dfrac{\sin^2(k'L)}{4(E/V)(E/V-1)}}, \qquad \hbar k'=\sqrt{2mV(E/V-1)}. \]

The transmission coefficient fluctuates between maxima and minima as the energy increases. It reaches its maximum value of 1, making the barrier transparent, whenever \( \sin(k'L)=0 \), that is at \( k'=n\pi/L \) for \( n=1,2,\ldots \), and has minima near \( k'=n\pi/2L \) for odd \( n \). At high energies \( E \gg V \), \( T \) approaches its classical value of 1<sup>[4](https://web.stanford.edu/~oas/SI/QM/Atkins05.pdf)</sup>. These transparency resonances occur whenever the de Broglie wavelength inside the barrier resonates with the barrier length, \( \sin k_0 a = 0 \)<sup>[2](https://people.umass.edu/bvs/614_1D.pdf)</sup>.

At exactly \( E=V_0 \), taking the limit of either expression gives \( T=1/(1+\xi^2/4) \), so for a wide barrier the transmission remains far below unity just above the energy where a classical particle would move freely over the barrier<sup>[2](https://people.umass.edu/bvs/614_1D.pdf)</sup>.

## Extensions and approximations

**Transfer matrix method.** The rectangular barrier solution can be packaged into a transfer matrix relating the wavefunction coefficients on one side to those on the other, with \( T=1/|t_{11}|^2 \). Chaining propagation and discontinuity matrices treats a smooth barrier, such as the [Coulomb barrier](https://www.edgepedia.org/coulomb-barrier) in the alpha decay of \( ^{212}\mathrm{Po} \), as a sequence of adjacent rectangular barriers, an approximation that improves as the number of segments increases<sup>[6](https://facultystaff.richmond.edu/~ggilfoyl/research/gilfoyle/node2.html)</sup>. The same transfer-matrix approach underlies analyses of electron tunneling in semiconductor heterostructures and double-barrier resonant-tunneling devices<sup>[8](https://www.sciencedirect.com/topics/mathematics/transfer-matrix)</sup>.

**WKB approximation.** For a general barrier the Wentzel–Kramers–Brillouin (WKB) method gives \( T \approx \exp\left(-2\int_{x_1}^{x_2}|k(x')|\,dx'\right) \) between the classical turning points \( x_1 \) and \( x_2 \). For a constant-height barrier this integral reduces to \( 2\kappa L \), reproducing the rectangular-barrier exponential. The WKB criterion for validity implies the transmission probability is very small, so the approximation applies only when tunneling is rare<sup>[9](https://farside.ph.utexas.edu/teaching/qmech/lectures/node49.html)</sup>.

## Applications

**Field emission and the STM.** A strong external electric field of magnitude \( E_g \) normal to a metal surface tilts the surface potential barrier, giving an electron of kinetic energy \( E \) a barrier of width \( L = \varphi/(eE_g) \), where \( \varphi \) is the work function. Because \( L \) is inversely proportional to the field strength, raising the field makes the barrier narrower and the tunneling probability exponentially larger; the resulting tunneling-electron current is proportional to that probability<sup>[5](https://openstax.org/books/university-physics-volume-3/pages/7-6-the-quantum-tunneling-of-particles-through-potential-barriers)</sup>. This is the operating principle of the scanning tunneling microscope (STM), invented in 1981 by [Gerd Binnig](https://www.edgepedia.org/gerd-binnig) and [Heinrich Rohrer](https://www.edgepedia.org/heinrich-rohrer). Its tungsten, platinum-iridium, or gold tip is held 0.4 to 0.7 nm above the sample by a piezoelectric device, and measuring the tunneling current at each position yields a resolution of about 0.001 nm, roughly 1% of an atom's average diameter, enough to image individual atoms<sup>[5](https://openstax.org/books/university-physics-volume-3/pages/7-6-the-quantum-tunneling-of-particles-through-potential-barriers)</sup>.

**Nuclear alpha decay.** The escape of an alpha particle from a nucleus is treated with the same machinery: the smooth Coulomb potential \( V(x)=Z_1Z_2e^2/x \) between the nuclear radius and the classical turning point is divided into a sequence of adjacent rectangular barriers, and the transmission coefficient is extracted from the product of their transfer matrices<sup>[6](https://facultystaff.richmond.edu/~ggilfoyl/research/gilfoyle/node2.html)</sup>.

**Resonant tunneling devices.** Stacking two rectangular barriers with a well between them produces resonant tunneling, in which transmission peaks when the incident energy matches an eigenstate of the well; this effect is exploited in resonant-tunneling diodes and analyzed with transfer-matrix methods in GaAs/GaAlAs heterostructures<sup>[8](https://www.sciencedirect.com/topics/mathematics/transfer-matrix)</sup><sup> • </sup><sup>[10](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.45.3461)</sup>.

## References

1. [Quantum Tunneling through Potential Barriers, Physics Book (Georgia Tech)](https://www.physicsbook.gatech.edu/Quantum_Tunneling_through_Potential_Barriers)
2. [Lectures on Quantum Mechanics, Example 3: Rectangular barrier (University of Massachusetts Amherst course notes)](https://people.umass.edu/bvs/614_1D.pdf)
3. [Rectangular potential barrier, Bhatter College e-Learning notes (Semester IV Honours)](https://bhattercollege.ac.in/old/E_Learning/Physics/Rectangular_potential_barrier_Sem_IV%28H%29_CC9T_05_04_2020.pdf)
4. [Atkins, Molecular Quantum Mechanics, §2.14–2.15 (Stanford course text)](https://web.stanford.edu/~oas/SI/QM/Atkins05.pdf)
5. [University Physics Volume 3, §7.6: The Quantum Tunneling of Particles through Potential Barriers, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/7-6-the-quantum-tunneling-of-particles-through-potential-barriers)
6. [The Transfer Matrix Method, Gilfoyle (University of Richmond)](https://facultystaff.richmond.edu/~ggilfoyl/research/gilfoyle/node2.html)
7. [Physics 227 Chapter 4, University of Washington virtual book](https://faculty.washington.edu/seattle/physics227/reading/ch4-virtual-book.pdf)
8. [Transfer Matrix, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/mathematics/transfer-matrix)
9. [WKB Approximation, Farside Quantum Mechanics lectures (University of Texas at Austin)](https://farside.ph.utexas.edu/teaching/qmech/lectures/node49.html)
10. [Transfer-matrix theory of energy levels and electron tunneling in heterostructures, Phys. Rev. B 45, 3461](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.45.3461)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum phenomena and measurement › Quantum tunnelling › Tunnelling theory and approximation methods*

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