# 2.2.2 Energy levels

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Glossary -
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In this section:

## 2.2.2 Energy levels

Energy levels in atoms are known to be discreet rather than forming
a continuous set. Light emission from a hot hydrogen gas therefore
yields a spectrum consisting of individual lines at specific
wavelengths rather than a continuous
distribution of wavelengths. Understanding this discreet nature of the energy levels
and the calculation of the energies requires the use of quantum mechanics
as classical mechanics can not describe atomic systems correctly.
In this section we will present the details of two particular quantum systems:
That of a particle in a one-dimensional potential well and that of
an electron surrounding a proton as is the case in a hydrogen atom. For a more in depth analysis we refer the reader to
the
reference texts.

### 2.2.2.1 The infinite quantum well

The infinite well represents one of the simplest quantum mechanical
problems: it consists of a particle in a well which is defined by a
zero potential between *x=0* and
*x=L*_{x} and an infinite potential on
either side of the well. The potential and the first five energy
levels are shown in the figure below:

qwwell.gif

**Fig.2.2.1** *Potential energy of an infinite well with
width ***L**_{x}. Also indicated are the lowest five energy levels
in the well.

The energy levels in such a infinite well
are given by:
(f31)

where *h* is Planck's constant and *m*^{*} is the effective
mass of the particle. *n* is the quantum number associated with
the nth energy level, with energy *E*_{n}. Note that the lowest possible energy is not zero
even though the potential is zero within the well. Also that the distance
between adjacent energy levels increases as the energy increases. Two electrons with
opposite spin can occupy each level as *n* and *s* are
the only two quantum numbers needed to describe this system.

qwwell.xls - qwwell1.gif - qwwell2.gif

**Fig.2.2.2** *Energy levels, wavefunctions (left) and probability
density functions (right) in an
infinite quantum well. The figure is calculated for a 10 nm
wide well containing an electron with mass ***m**_{0}.

### 2.2.2.2 The finite quantum well

The potential and the first three energy
levels are shown in the figure below:

sqwell4.gif

**Fig.2.2.3** *Potential energy of a finite well with
width ***L**_{x}. Also indicated are the lowest three energy levels
in the well.

sqwell.xls - sqwell2.gif - sqwell3.gif

**Fig.2.2.4** *Energy levels, wavefunctions (left) and probability
density functions (right) in a finite quantum well.*

Example - Problem sspp006

### 2.2.2.3 The hydrogen atom

The hydrogen atom represents the simplest atom since it consists of
only one proton and one electron. The potential is due to the
electrostatic
force between the positively charged proton and the negatively
charged electron. This potential as well as the first three
probability density functions
(*r*^{2}|Y|^{2})
of the radially symmetric wavefunctions
are shown in the figure below.

hydrog.gif

**Fig.2.2.5** *Potential energy (black curve) in a hydrogen
atom. Also shown are the probability densities for
***n=1 and l=0** (red curve), **n=2 and l=0** (green curve) and
**n=3 and l=0** (blue curve). The probability densities
are shifted by the corresponding electron energy.

The possible energy levels of the electron in the hydrogen atom
are given by:
(f32)

where *m*_{0} is the reduced mass of the electron and *n* is
the primary quantum number.

hydrogen.xls - hydrogen.gif

**Fig.2.2.5** *Energy levels and possible electronic transitions
in a hydrogen atom. Shown are the first six energy levels
corresponding to ***n = 1, 2, 3, 4, 5, and 6** as well
as the possible transitions involving the lowest energy level
(**n = 1**). The transitions are shown as red arrows and
represent the first five transitions of the **Lyman** series.
The blue arrow indicates the largest possible energy difference between
two bound states and equals 13.6 eV (sometimes called one Rydberg).

There are three more quantum numbers needed
to describe this system fully, namely *m*, *l* and *s*. These
quantum numbers do determine how many quantum states have the
same energy. Note that all the possible energy values are negative.
Electrons with a positive energy are not bound to the proton and behave
as free electrons.

2.2.1
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2.2.3

© Bart J. Van Zeghbroeck, 1996, 1997