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In 1926, An Austrian physicist

In 1926, An Austrian physicist

In 1926, an Austrian physicist by the name of Erwin Schrödinger developed an equation known as the Schrödinger Wave Equation.  What’s interesting about this equation is that it brings into light the idea of de Broglie about the electron behaving like a wave and like a particle.  The Schrödinger wave equation incorporates both perspectives.  In reality, the Schrödinger equation requires advanced calculus (and truthfully while I have had the math classes for it, I have yet to take the course that have the applications of math directed towards quantum mechanics), so I’ll focus on the qualitative nature of QM.  The solving of the Schrödinger equation will lead to a series of functions referred to as wave functions.  These wave functions are mathematical in nature and are denoted by the Greek letter psi ().  It is important to note that while the wave function itself does not directly reveal any information, 2 (the square of the wave functions) provides information about an electron’s location in a permissible energy state.  Bohr’s model of the hydrogen atom has the assumption that the orbit pathway of an electron is circular in nature.  However this description cannot be valid in quantum mechanics.  To add to the difficulties of determining the position of an electron, the Uncertainty Principle prohibits us from knowing the whereabouts of an electron (assuming we are aware of the momentum).  Because of this, we cannot hope to single out the physical location of an individual electron.  Due to this vague nature, the proper term to use in describing the location of an electron is probability.  Concerning the (Schrödinger wave function)2, 2 really represents the probability of finding an electron in that area.  Because of this, 2 is referred to as the probability density.  Regions with a high probability of finding an electron correspond to high values for 2.

And the Quantum Numbers?

For the hydrogen atom, the complete solution for the Schrödinger equation reveals a set of wave functions and a set of energies that correspond.  The wave functions in this case are known as orbitals (note that we use the QM model orbital to differentiate from Bohr’s orbit).  Each orbital will describe a distribution of electron density as given by the probability density.  In quantum mechanics, there are four distinctive numbers that are used to describe an individual electron.

The primary number brought up is called the principle quantum number (n).  This number can have any integral value from 1 to infinity.  As n increases, the shell becomes larger and the electron spends a greater amount of time far from the nucleus.  Higher values of ncorrespond with higher energies and therefore implying that the electron is not as tightly bound to the nucleus.  Essentially, this number identifies the shell (or level) to which a particular electron belongs.  The maximum number of electrons that a shell can contain can be demonstrated as a function of n (emax = 2n2).

Each shell will consist of one or more subshell (or sublevels).  The number of subshells in a principal shell is equal to the number n (i.e. there is one subshell in the n = 1 shell).  Each subshell within a shell is designated what is called a subsidiary quantum number (l).  The values of l are determined by values of n.  The value l can equal any number from 0 to (n – 1).  So, using this scheme, when n = 1(its lowest possible value) then l = 0.  To avoid the confusion that the two numbers would bring, scientists used a letter to represent each value of l.  For instance l = 0, 1, 2, 3, 4… and so the notation would be s, p, d, f, g….  The s and p values tend to throw the whole scheme out of place but for l values higher than 3, the letters will proceed in an alphabetical order.  In order to designate a subshell, one combines a principal quantum number with a letter equivalent value of l.  For example, the subshell with n = 3 and l = 1 is called the 3p subshell.  Neat huh?  Each of these subshells will consist of at least one orbital.  The number of orbitals is equal to two times the l value plus 1                (number of orbitals = 2l + 1).

As mentioned above, there is at least one orbital within a subshell.  Each orbital is designated and identified by a magnetic orbital quantum number (ml).  This number can have the numerical range from –l to +l.  So for an l value of 0, the only permitted value of ml is 0.  For l = 3, the only permitted ml values are –3, –2, -1, 0, +1, +2, +3 (which are seven f orbitals).  The ml value of an orbital is used to describe the orientation of the orbital in space.  The maximum number of electrons for a subshell can be calculated by multiplying the number of orbitals in the subshell by two.

The fourth quantum number, the magnetic spin quantum number (ms), is absolutely necessary to describe the electron completely.  This number arises due to the fact that the electron possesses magnetic properties that are similar to the properties possessed by a charged particle spinning on its axis.  The spinning generates a magnetic field, which can be associated with a magnetic spin (hence the name).  The value is very simple to remember since it involves only two values [-(1/2) and +(1/2)].  The values indicate that the electrons have opposed spins.  One concept of extreme importance is the Pauli Exclusion Principle (devised by the Austrian physicist, Wolfgang Pauli).  The exclusion principle states that no two electrons in the same atom may have the same set of quantum numbers.  Because of this, an orbital may hold no more than two electrons (only two numerical options for magnetic spin).