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Quantum Tunneling

Main Concept

In the world of classical physics, a particle with energy E cannot pass through a potential well of height V0  if E<V0. Yet in quantum mechanics, a particle can access regions where it is classically forbidden even though E<V0. This phenomenon is called quantum tunneling, and it is responsible for the functionality of the transistors which are used to make computers. Because the particle's wavefunction must be continuous everywhere, the probability of finding the particle cannot instantly vanish when E<V0, instead it exponentially decays in these regions. Quantum tunneling also plays a critical role in nuclear fusion, electronics and quantum computation. In this document, you can derive and animate the effects of a stationary wavefunction tunneling through a potential barrier.


Derivation of time-independent Schrödinger equation


The time-dependent Schrödinger equation for the wavefunction &Psi;r&comma;t is:


i &hslash; &Psi;t&equals;H ˆ&Psi;,

where H ˆ is the Hamiltonian operator, and  is the reduced Planck's constant. For the case of a single non-relativistic particle, the Hamiltonian takes the form &hslash;22 m2&plus;Vr&comma;t:


i &hslash; &Psi;t&equals;22 m2&plus;Vr&comma;t &Psi;,

where m is the mass of the particle, V  is its potential energy, and 2 is the Laplacian. Simplifying to one dimension, and assuming the potential is constant in time, you get:

i &hslash; &Psi;t&equals;22 m2x2&plus;Vx &Psi;.


Recalling that the modulus squared of the wavefunction, &Psi;x&comma; t2, is the probability density of the particle, a stationary state is a state where this probability density does not change in time, that is, the particle stays in the same state in every observable way. As such, you observe that &Psi;x&comma; t must take the form:


Ψx&comma; t &equals; &psi;x &ExponentialE;i &theta;x&comma;t,


for some real valued function &theta;x&comma;t. Including further restriction that the potential barrier Vx is a well-behaved function with no vertical asymptotes, it can be shown that &theta;x&comma;t = E&hslash; t, where E is the energy of the particle. Plugging this solution into the time-dependant Schrödinger equation produces the time-independent Schrödinger equation:


 &hslash;22 m 2x2&plus; Vx  &psi;x&equals;E &psi;x.

or simply H ˆ&psi;&equals;E &psi;.


Derivation of tunneling


Consider the one-dimensional time-independent Schrödinger equation for a piecewise constant potential Vx.

Assume that Region 1 of space  on the interval &comma; 0 has potential energy V1. Region 2 has potential energy V2 and is on the interval 0&comma; a&comma; and lastly, Region 3 has a potential energy V3 and is on the interval a&comma; . The total potential can be mathematically written as:

 Vx &equals; &lcub;V1&comma;x0&comma;V2&comma;0<x  a&comma;V3&comma;a<x&comma;   


This is a very general derivation because you can vary the height of any potential well along with the central width a. The time-independent Schrödinger equation must be solved in the three regions and the solutions connected by junction conditions, that is, the requirement that the wavefunction and its derivative be continuous on the boundaries. If you the call the three solutions &psi;1&comma;&psi;2&comma;&psi;3 respectively, then the junction conditions are:


 ψ10&equals; ψ20&comma;ψ2a &equals; ψ3a&comma;  &psi;&apos;10&equals; &psi;&apos;20&comma; &psi;&apos;2a &equals; &psi;&apos;3a&comma;  


where the primes denotes differentiation with respect to x. The solutions to the Schrödinger equation for E&gt;V1 in these three regions are:


ψ1x &equals; Aeik1x&plus;Beik1x&comma; &psi;2 x &equals; Ceik2x&plus;Deik2x&comma;ψ3x &equals; Feik3x&plus;Geik3x&comma;



k1&equals; 2mEV1&hslash;2&comma; k2&equals; 2mEV2&hslash;2&comma;k3&equals; 2mEV3&hslash;2&period; 


Notice that if E&gt;V, the wavefunction is a complex plane wave with the form ei k x with real-valued k, while if E<V, the form of the solution is ek x with a real-valued k. The case of E&equals;V  has a different solution and is not considered here.


To extract more physically realizable quantities, it is necessary to assume that the particle or wavefunction 'originates' from only one side of the potential barrier. Physically, this is because of the assumption that there is no source of particles on the right hand side which travel in the x direction. This makes the constant G&equals;0, since the function Geik3xis a plane wave traveling in the x direction.


Now by applying the junction conditions above, you can derive the constants B, C, D, F, in terms of A, the amplitude of the wave. This can be very messy, especially in our highly general case of arbitrary V1&comma; V2&comma; V3&period; To extract physically meaningful quantities from these abstract functions, the reflection and transmission coefficients are defined as the ratio:

R &equals; B2&verbar;A&verbar;2&comma;T &equals; F2A2&comma;


so that R &plus; T &equals; 1. This gives the amount of the probability density that is "reflected" or "transmitted", similar to that used in optics. Remember that this is a solution for the time independent Schrödinger equation, so the particle is in a stationary state. It is not traveling through the potential barrier in time, rather it leaks through the barrier as a result of the plane wave solution to the Schrödinger equation.


Use the sliders to adjust the shape of the potential barrier, and the energy and amplitude of the incoming wave. You can also stop or play the animation using the checkbox, or move through time manually with the slider. Use the checkboxes above the plot to show different components of the wavefunction.



What happens when you make the potential width very thick?


Try adjusting the height of the potentials.


Try plotting the real, imaginary, or square of the wavefunction.



Barrier Width,  a

Initial Potential,  V1

Potential Barrier,  V2

Final Potential,  V3  

Energy,  E

Amplitude,  A

Time,  t

Transmission Coefficient,  T

Reflection Coefficient,  R

                                   Wavefunction components



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