# 2.2.2 Energy levels

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=Lx 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 Lx. 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 En. 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 m0.

### 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 Lx. 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 (r2|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 m0 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 ¬ ­ ® 2.2.3

© Bart J. Van Zeghbroeck, 1996, 1997