
Quantum Wave Mechanics – The Schrödinger Equation 

2.1 The timedependent Schrödinger equationClassical mechanics uses deterministic equations to define the state and motion of a macroscopic system at a given point. According to Newton’s second law (equation (2.1)), the future state and motion of a system can be exactly determined, given the state of the system at any time  an example of this is given in Levine.[7]
For a microscopic particle, Heisenberg’s
uncertainty principle shows that we cannot know the exact position
and velocity of a particle at a specific simultaneous point in time –
this means classical mechanics cannot be used to predict the state and
motion of microscopic – or quantum
– particles.
…where i represents the square root
of –1.
…where m is the mass of the particle, V(x,t) is the potential energy function of the system, i again represents the square root of –1, and the constant ħ is defined as in equation (2.4):
Equation (2.3) is known as the timedependent
Schrödinger (wave) equation. It represented the start of Schrödinger’s
work into quantum wave mechanics, most notably for proving that matrix
mechanics (treating the electron as a quantum particle, the work of Heisenberg)
and wave mechanics (treating the electron as a wave) were compatible.
He was jointly awarded the Nobel
Prize for Physics in 1933 with Paul
Dirac for this work (Dirac for his work on electron and positron theory).
…where Ψ* represents the complex
conjugate of Ψ. 2.2 The timeindependent Schrödinger equationIn chemistry, the simpler timeindependent Schrödinger equation is more often used than the timedependent Schrödinger (equation ((2.3)). This is derived from the timedependent version by considering the special case where the potential energy, V, is a function of only position, x and not of time, t. Solutions of (2.3) that can be written as a product of a function of time and a function of x:
Partial derivation of (2.6) gives:
Which, upon substitution into the timedependent Schrödinger equation, (2.3), gives:
…which is divided by fΨ:
Each side of (2.9) is expected to be a function of x and of t. The right side, however, does not depend on t, and so is independent of it. Similarly, the left side is independent of x. Since the function is independent of both variables, x and t, it must be a constant. This is called E. Equating the right side of (2.9) to E gives:
This is the timeindependent Schrödinger
equation for a single particle of mass m
moving in one dimension. E has the same units
as V, and is therefore postulated to be the
energy of the system.
This then means that where the potential energy is a function of x only, wavefunctions of the following form exist:
The probability density function can be shown to be timeindependent thus:
2.3 The threedimensional manyparticle Schrödinger equationThe timeindependent Schrödinger equation is often represented as energy eigenfunctions and eigenvalues, using the Hamiltonian operator, Ĥ:
For a oneparticle, threedimensional system, the classicalmechanical Hamiltonian is as given in equation (2.15):
Quantummechanically, this is represented as in equation (2.16):
…where the section in parentheses is called the Laplacian operator, Ñ ^{2} (equation (2.17)).
The threedimensional, oneparticle timeindependent Schrödinger equation is thus represented as in equation (2.18):
A many particle system can be represented by considering a system of
n particles, where particle i
has mass m_{i} and spatial coordinates
(x_{i}, y_{i},
z_{i}), where i=1,2,3,…,n.
The potential energy is usually only taken to be dependent on the 3n spatial coordinates. The timeindependent Schrödinger equation then becomes:


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[8] E. Schrödinger, “Quantisierung als Eigenwertproblem”, Ann. d. Physik, 79, 489, 1926  80, 437, 1926  81, 109, 1926 
Potential Energy Surfaces and Conical Intersections • June 2002 • Ian Grant 