Wave Functions Of A Particle Trapped Within A Box

The standing wave formed for a particle trapped within a box is analogous to the standing wave formed on a string stretched between two rigid supports. It can thus be deduced that the wave function ψn(x) of this particle has the same form as the displacement function yn(x) for the standing wave on a string stretched between the two rigid …

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Correspondence Principle

The quantization of energy, the zero point energy and the uncertainty principle should all appear strange and against “common sense”. All of these results and many others predicted by quantum mechanics have been exhaustively tested experimentally. In fact, quantum mechanics gives the correct results for objects of all sizes. The “strange” effects referred to above are simply too small for …

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Allowed Energy States Of A Trapped Particle

Using de Broyglie’s equation $p = \frac{h}{\lambda}$ and $L = n \frac{\lambda_{n}}{2}$, the momentum of the particle in its nth mode is given by: $p_{n} = \frac{h}{\lambda_{n}} = \frac{nh}{2L}$ The momentum p is also related to the kinetic energy of the particle Ek by: (To derive this formula, sub. p = mv into KE eqn) $E_{k} = \frac{p^{2}}{2m}$ Since the …

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Particle Trapped In A Box Of Length L

Consider a particle (e.g. an electron) that is confined to a “1-dimensional box” of length L. A 1-dimensional box is taken to be one in which its potential energy V = ∞ outside the box. Inside the box, no force acts on the particle and so its potential energy is constant which can be fixed arbitrary to be zero, ie …

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Scanning Tunneling Microscope

The scanning tunnelling microsope (STM) is a non-optical microscope which uses the concept of quantum tunnelling by electrons to study surfaces of conductors or semi-conducors at the atomic scale of about 2 Å or 0.2 nm.   Steps: An atomically sharp probe (the tip) is moved over the surface of the material under study The electrons at the tip of …

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Reflection And Transmission

  In the above diagram, the oscillating curve on the left of the potential barrier is a standing wave pattern that results from the interference between the incident matter wave and the reflected matter wave that has a smaller amplitude than the incident matter wave. These two waves, of the same type, travelling in opposite directions interfere with each other …

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Quantum Tunnelling

Potential Barrier Consider an electron travelling along an ideal wire whose electric potential varies as shown below.   The negatively charged electron, which possesses an initial energy E, approaches the region (or ‘barrier’) of the wire, of width L, with negative potential. The electron is expected to gain electric potential energy of Uo = qv’ as it passes the region …

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