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Schrodinger model

The Schrödinger model of quantum physics is an extension of the wave-particle duality concept.

In the Schrödinger model, a particle (e.g. electron) is represented as a wave function (denoted by the symbol $$\psi$$).

A wave function is a type of stationary wave that contains all of the information (e.g. the probabilities of position, velocity, energy) about the particle.

The absolute value of the wave function squared (i.e. $$\vert\psi\vert^{2}$$) for a given point in space is the probability density. The probability density gives the probability of finding a particle at a point.

The probabilistic nature of a wave function incorporates Heisenberg's uncertainty principle as it implies that the physical properties of a quantum object cannot be determined exactly.

A potential barrier is a region in space where the potential energy of a particle is higher than its surroundings.

The minimum energy required for a particle to pass through this barrier (in classical physics) is called the barrier height. It appears like a block in the middle of a graph of energy vs position.

In classical physics, a particle with a total energy which is less than the barrier height would be unable to cross the potential barrier.

For example, an object with insufficient kinetic energy cannot move infinitely away from the centre of a gravitational field. And a ball cannot roll to the top of a hill unless it has sufficient kinetic energy.

Quantum tunnelling describes the phenomenon in quantum physics that a particle lacking energy equal to the barrier height can still be found on the other side of the barrier.

The principle of quantum tunnelling is best explained using the Schrödinger model of quantum physics.

The model interprets the wave function of a particle as the probability of the particle being at a given point in space. The function is non-zero for any point at a finite distance.

We assume that a particle P on side A of a potential barrier has kinetic energy that is below the barrier height. If one calculates the value of the wave function (i.e. amplitude) for points on the other side of the barrier, the values will be very low but non-zero.

This means that the probability that the particle is on side B of the barrier is also non-zero. When this probability is realised (the particle actually is on the other side), tunnelling has occurred.

The wave function of the particle after tunnelling has a lower amplitude due to damping by the potential barrier.

If, by contrast, the particle has sufficient kinetic energy to pass from A to B (not by tunnelling), the wave is able to translate from A to B without any loss in amplitude.

A scanning tunnelling microscope (STM) is a device used to obtain atomic-scale images of a conducting or semi-conducting surface. It is an application of quantum tunnelling.

An STM has a probe with a sharp conducting tip, which usually ends in a single atom.

When scanning a surface, the distance between the probing tip and the surface is kept very small. As a result, electrons of the tip and the surface (of a high enough energy) can tunnel through the space between them (which forms a potential barrier).

A small potential difference applied across the barrier will cause a tunnelling current to flow across it. This current decreases exponentially with the tip-surface distance and hence allows the distance to be controlled very precisely.

The STM can be operated in two different modes, the constant current and constant height modes.

The constant current mode maintains the current across the barrier by maintaining the tip-surface distance. The tip follows the contour of the surface and the variation of the tip height is recorded by computers which generate an image of the surface.

The constant height mode maintains the vertical position of the tip and the variation of current is recorded by computers. The variation of the tip-surface distance is then determined from the variation of current and an image of the surface is generated.

The transmission coefficient $$(T)$$ is the probability that a particle tunnels through a potential barrier. It is approximated by: \begin{gather*} T=e^{-2kd}\\ k=\sqrt{\frac{8\pi^{2}(U-E)}{h^{2}}} \end{gather*} $$d$$ is the width of the potential barrier (in $$\text{m}$$), $$U$$ is the height of the potential barrier (in $$\text{J}$$) and $$E$$ is the total energy of the particle (in $$\text{J}$$).

The reflection coefficient $$(R)$$ is the probability that a particle is reflected by a potential barrier. Since the particle is either reflected by or transmitted through the barrier, $$R+T=1$$ (recall that the sum of all individual probabilities is equal to 1).