Page 235 - Instant notes

The wave nature of matter 221

Quantum tunneling is the probability of observing a particle
beyond a (non-infinite) potential energy barrier that exceeds the
energy of the particle. The effect arises because the amplitude of
a wavefunction

decreases exponentially within the wavefunction amplitude beyond barrier
resulting in non-zero the barrier.
Related Quantization of energy and particle-wave Many-electron atoms (G6)
topics duality (G3)
General features of spectroscopy
(I1)
The structure of the hydrogen atom (G5)

Wavefunctions and probabilities

In the particle-wave duality interpretation of matter and radiation (Topic G3) a particle
moving in space can also be described as a wave in space with a wavelength related to
the particle momentum by de Broglie’s equation, λ=h/p. In quantum mechanics, the
notion of a particle moving in defined trajectories in a system is replaced entirely by this
description of the system in terms of its wavefunctions, ψ. The wavefunction
simultaneously describes all regions of space in which the particle it represents can be
found. This, in turn, introduces the idea of uncertainty into quantum mechanics because
the exact position of the particle at each point in time is not defined, only the region of
space of all its possible positions. The exact shape of the wavefunction is important
2
because the probability of finding the particle at each point is proportional to ψ at that
point; a greater amplitude in the wavefunction corresponds to a greater probability
density in the particle’s distribution.

Schrödinger equation

The Schrödinger equation is the fundamental equation of quantum mechanics and has
the general form:
Hψ i=E iψ i

Each allowed wavefunction ψ 1, ψ 2, ψ 3…of a system described by a Hamiltonian
operator, H, is associated with one particular allowed energy level E 1, E 2, E 3… (An
operator is a mathematical function that represents the action of a physical observable.)
The Hamiltonian operator is the operator for the total kinetic and potential energy of the
system. Only an allowed wavefunction of the system, ψ, when operated on by H, returns
the same wavefunction, multiplied by the associated constant value E. In mathematical
terminology the Schrödinger equation is an eigenvalue equation; the pairs of E and ψ that
satisfy the equation are the eigenvalues and eigenfunctions of H, respectively.