Page 59 - Science at the nanoscale

RPS: PSP0007 - Science-at-Nanoscale
9:2
June 9, 2009
3.3. Hydrogen-like Atoms: Orbitals and Atomic Structures
If κW ≫ 1, then we can use the approximation sinh(κW)
1
exp(κW), and the transmission coefficient becomes
∼
2
16E(V o − E)
−2κW
e
(3.54)
T ≈
2
V
o
The probability that the particle can tunnel through the barrier
thus depends on the barrier width and the barrier potential height.
This relation is an important result for quantum tunneling and the
scanning tunneling microscope (see Chapter 8).
HYDROGEN-LIKE ATOMS: ORBITALS AND
3.3
ATOMIC STRUCTURES
In this section, we shall discuss the properties of atoms and ions
These atoms or ions are known as
having just one electron.
hydrogen-like atoms. The atom consists of a positively charged
nucleus with a charge of +Ze while a single electron (charge −e)
moves around the nucleus. Here Z corresponds to the number of
protons in the nucleus. Assuming that the nucleus behaves like a
point charge, the potential energy of such a hydrogen-like atom is
given by
2
Ze
(3.55)
V = −
4πε o r
where r refers to the separation between the nucleus and the
electron, and ε o is the permittivity of free space. To determine
the properties of the electron using quantum mechanics, we are
required to solve the Schr¨odinger equation for the hydrogen-like 49 ch03
atom using Eq. (3.55) for the potential energy. Bearing in mind
that the electron moves in all three dimensions, thus we have the
following Schr¨odinger equation for hydrogen-like atoms
2
2
2
h 2 d ψ d ψ d ψ Ze 2
− 2 2 + 2 + 2 − ψ = Eψ (3.56)
8mπ dx dy dz 4πε o r
The next task is to solve this equation for the wavefunction and
the energy of the system. The solution to Eq. (3.56) is rather com-
plicated, so instead of detailing the complete solution, we shall
outline some of the important properties of the equation and its
solutions.
Upon solving the above Schr¨odinger equation, we obtain the
following energy equation for the different states of the electron