# Why wavefunction becomes exponentially smaller during quantum tunneling?

I am interested in quantum tunneling and I am wondering why the wavefunction of a particle would becomes smaller so that there is a slight possibility of finding it at the other side of a big energy barrier? Is there any interaction otherwise how can the wavefunction knows there is a barrier?

• I mean, the presence of a potential barrier is what defines an interaction between the object and some container, so... yes? – probably_someone May 10 at 13:16
• Do you want a "why" answer other than the fact that that's the solution of the Schrodinger equation? Do you want it at a lower mathematical level than that? – Ben Crowell May 10 at 15:10

## 3 Answers

Wavefunctions are solutions of quantum mechanical differential equations, with given boundary conditions for the problem at hand, i.e. tunneling: Is there any interaction otherwise how can the wavefunction knows there is a barrier?

The boundary condition of a barrier defines the wavefunction by construction. It has been found that experiments validate this model and its predictions.

• This answer does not really address the OP question directly. Where have they misunderstood "interaction"? What about the case when there are no boundary conditions, e.g. a smooth potential hump? – ggcg May 10 at 14:23

The classic step barrier is the easiest to solve but the general behavior holds for other functions. The comment addresses the physics of your question, i.e. "the presence of a potential barrier is what defines an interaction". This is the same for classical mechanics. One could ask, "How does the earth know to move in an ellipse when gravity is there, is there an interaction"?

Specific details come from the math. Schrodinger's equation is set up in each interval of the x-axis (assuming a 1-dim problem) with the appropriate value of V(x) in each region. If the total energy is greater than the barrier there will be a modification of the wavelength in the region of the barrier. If the total energy of the particle is less than the barrier then the wave number will become imaginary, cancelling the imaginary factor, i, in the exponential leading to a decaying solution. If the barrier continues to exist for all space after it is encountered the decay will eventually cause the wave function to vanish, this is the phenomenon of penetration. If the barrier is finite in extent then the particle will make it out the other side, tunneling. It is fairly easy to derive the exact solution for this case since the potential is constant, they are exp(+ikx) and exp(-ikx), with appropriate boundary conditions.

• So what's wrong with this answer? – ggcg May 10 at 14:21
• Nothing wrong, it is a correct description of the mathematical model of tunneling. – anna v May 11 at 2:59

"Is there any interaction otherwise how can the wavefunction know there is a barrier?"

The term interaction means that two systems are influencing one another. It means they are exerting a force on each other. In quantum theory that force is most conveniently expressed by writing down the potential energy. That potential energy is the thing you are calling a barrier. Therefore the potential energy barrier is the interaction.

Two examples: 1. an electron moving between a pair of grids to which electrical voltages are applied. 2. an alpha particle emerging from a nucleus undergoing alpha decay. In the first the charge interacts with the electric field and this results in potential energy; in the second the particles interact via the strong interaction and this results in potential energy, and there is also an electromagnetic part. These potentials provide the barrier.

If you want to know why or how the entity (call it a particle if you like, but you could equally call it a wave) gets through the barrier, you should perhaps also ask yourself why it would not. The only way to answer is to examine the physical description of the evolution of the wave/particle. That description is provided by the Schrodinger equation, which tells how the wavelength is affected by the kinetic energy and things like that. Basically when waves undergo reflection they generally extend a short way into the reflecting region, with a characteristic decay length, and if the reflecting region is less thick than this decay length then there will be partial transmission and less than complete reflection. This happens also with ordinary waves such as waves on water, or light waves or whatever you like. The reason why people find it more interesting in the quantum case is because the thing that gets partially transmitted also has particle-like characterstics, such as when it interacts with another thing later on in such a way as to transfer its energy.