In the last lecture, we saw that the Bohr model is able to predict the allowed energies of any single-electron atom or cation. However, the Bohr model is, by no means, a general approach and, in any case, it relies heavily on classical ideas, clumsily grafting quantization onto an essentially classical picture, and therefore, provides no real insights into the true quantum nature of the atom.

Any rule that might be capable of predicting the allowed energies
of a quantum system must also account for the particle-wave
duality and include a wave-like description for particles.
In 1926, the Austrian physicist Erwin Schrödinger posited
an equation that predicts both the allowed energies of
a system as well as the probability of finding a particle
in a given region of space. The equation, known as
the Schrödinger wave equation, does not yield the probability
directly, in fact, but rather the probability amplitude
alluded to in the previous lecture. This amplitude
function is, in general, a *complex* function
denoted (for a single particle in one
spatial dimension) and is referred to as
the *wave function*. It is related to the probability
as follows:

*The probability that a single quantum particle moving in one spatial dimension will be found in a region if a measurement of its location is performed is*

Since particles can exhibit wave-like behavior, the amplitude or wave function should have a wave-like form.

The Schrödinger equation cannot be derived from any more fundamental
principle. However, in order to motivate it, let us use the
assumption that should have a wave-like form.
Thus, consider a free particle of mass and momentum
. Recall the de Broglie hypothesis stating that the
particle has a wavelength given by

or

If the particle is a free particle, its potential energy , so that its energy is purely kinetic

If the amplitude describes a wave, then it should take the mathematical form

(we are considering a wave that is not changing in time here). Consider the cosine form (the same will hold for a sine for as well) and consider the first two derivatives of :

We see, therefore, that and are related by

or

The last line of the above expression is, in fact, the Schrödinger equation for a free particle moving along the -axis.

We can always set up the Schrödinger equation via the following simple
prescription. Start with an expression for the classical energy.
In this case, for a free particle

Now multiply by the wave function :

Finally replace the momentum by the following derivative:

which is equivalent to replacing by a second derivative:

When this is done, we arrive at:

If the particle has a potential energy affecting it, then the same prescription can be used. Start with the classical energy expression:

Multiply by :

Replace , and we arrive at the Schrödinger equation for the general case of a single quantum particle in one spatial dimension:

The object on the left that acts on

is an example of an

Therefore, the Schrödinger equation is generally written as

Note that is derived from the classical energy simply by replacing .

Since
is a probability, we require that the
probability of finding the particle *somewhere* in space
be exactly 1. That is, we require that the
probability that
be 1, which means

This is known as the

Finally, we need to specify how behaves at the physical boundaries
of the space we are working in. These conditions are known as
*boundary conditions*.

Once we specify the Schrödinger equation, the boundary conditions
on and the normalization condition, we have
all the information we need to calculate both the allowed
energies and the wave function . We will see shortly how
this prescription is applied to a few simple examples.

Before we embark on this, however, let us pause to comment on
the validity of quantum mechanics. Despite its weirdness,
its abstractness, and its strange view of the universe
as a place of randomness and unpredictability, quantum theory
has been subject to intense experimental scrutiny. It has
been found to agree with experiments to better than
10% for all cases studied so far. When
the Schrödinger equation is combined with a quantum description
of the electromagnetic field, a theory known as
*quantum electrodynamics*, the result is one of the
most accurate theories of matter that has ever been
put forth. Keeping this in mind, let us forge ahead
in our discussion of the quantum universe and how to
apply quantum theory to both model and real situations.