Bohr's model worked for a simple system, but did not completely explain additional detail associated with the emission spectrum. To a point it worked. In these circumstances modifications of the model are typical needed. The primary modification of the Bohr model which resulted in our current view of the electronic structure of the atom was to recognize the electron exhibited wave properties as well as particle properties. This rather simple statement was the result of considerable labor. Let's go back to the origin of the ideas which subsequently lead to this important modification of describing the behavior of an electron in an atom.

In his study of the Photoelectric effect Einstein recognized that light exhibited particle behavior. When light of specific frequency was incident on a metal surface electrons were ejected from the metal. The incident light in causing electrons to be ejected was behaving as a particle. Light, electromagnetic radiation, could not only be thought of as having wave properties, but also particle properties. A packet of light (photon) had a frequency, but it also had an energy given by;

Now we began paying a price for our search to elucidate the structure of the atom and a gentleman by the name of Louis Victor de Broglie is responsible. What he did was to make the atom an abstract and and difficult to imagine. He made the universe probabilistic and indeterminate. Now understand that the universe has not changed, but our view of it has.

The pivotal concept suggested by de Broglie, was that if light could exhibit particle properties, and therefore, help us explain phenomenon, then why could not matter have wave properties. de Broglie was saying that matter, under certain circumstances might also display appreciable wavelike properties. He suggested that the wavelength of matter behaving in this manner could be calculated using the relationship;

where 'm' is the mass of the particle and 'v' is its velocity.

Called matter waves they should not be confused with the waves of electromagnetic radiation. Matter waves are not radiated into space or emitted by a particle; they are never dissociated from the particle. The speed of matter waves is never the speed of light, nor is it constant, but changes with the particle. de Broglie suggested that electrons, because of their very small mass and high velocity, could behave as waves. There existed experimental evidence to support this idea. Even Thomson had observed the beam of electrons in the cathode ray tube would cast a shadow on the end of the tube when a piece of metal is placed in the beam.

For years after de Broglie's postulate Clinton Davidson and L. Germer investigating the scattering of electrons by atoms in a crystal showed that the electron beam was reflected (diffracted) in an identical way that X-rays were diffracted. Exhibiting the wave-like character of the electron. Erwin Schrodinger, an Austrian theoretical physicist, assumed that the wavelike character of the electron could be incorporated into a model of the atom. And here is where our world falls apart, and it becomes very difficult to describe electron behavior in atoms. Because now the medium has turned form pictures to probability and quantum mechanics or wave mechanics. The language is differential calculus, the words are symbols and numbers. The results are complex, but nonetheless dramatic.

We will avoid the pit and try to describe the results. Try is what I will do, because quantum mechanics changes the way we think about the motion of particles as small as the electron. Large objects moving slowly are easy to follow. Like a baseball. A propeller turning slowly is also easy to follow, However, as we speed up the blade it becomes difficult to know exactly where the two ends of the propeller are located. Their movement describes a region where, if we were to insert a pencil, we would expect to encounter one end of the blade or the other. Outside its length we would assume we would be safe. In the world of the atom the motion of the electron is equally complex. It can be thought of as smoothed out around the nucleus. We can not see an individual electron or how fast it is going.

Werner Heisenberg showed by using quantum mechanics that it is impossible to know both the position and velocity of the electron;

Now this problem is only encountered in the realm of the atom, we do not have such problem in the macroscopic world. (If we did more of us would pay closer attention to the 'walk' and 'don't walk' sign at traffic signals). Heisenberg's uncertainty principle says that because of the wavelike character of the very small electron we can not know precisely where the electron is when traveling along a path or trajectory.

If we accept that an electron has wave character than an electron can be described as a wave. If we use a single wave to describe the velocity of the electron, as shown in the animation, we do not know where the electron is located exactly. We have an idea of its vertical location, between the peak maxima, but we do not know where it is located horizontally. If we use a number of different waves and add them together we can better locate the position of the electron, but we have no idea what the velocity of the electron is because of the different velocities of the waves that contribute to the electron. So Heisenberg's uncertain principle concludes that we can not know both the position and the velocity of an electron simultaneously. This means we can not know exactly where the electron is an any instant.

We can not know how fast an electron moves in an atom (about the nucleus). However, quantum mechanics does allow us to describe the behavior of the atom in a statistical sense. We talk about the probability of finding the electron at a certain point in space in the vicinity of the atom. We can not describe the electron as orbiting the nucleus. Because to do so suggests the electron has a trajectory which suggests we can know both the position and velocity of the electron. So in quantum mechanics we describe the behavior of the electron in three-dimensional terms. Instead of finding an electron in an orbit, as Bohr's model suggested, we describe the electron in an orbital, which is a 3-dimensional orbit. But now the question becomes what 3-dimensional shape or shapes does an electron in a hydrogen atom have?

The information that we are seeking can be obtained by solving a complex mathematical relationship, referred to as Schrodinger's equation. It is a complex differential equation, the unknown in the equation is , a mathematical expression called a wavefunction. The square of is a measure of the relative probability density of the electron at a position (x,y,z). In optics the square of the amplitude of a wave is called the intensity of radiation. We also recall the intensity is interpreted as the number of photons present. Bright light suggests many photons. For wavefunctions the square of the amplitude indicates the probability of finding the particle at each point in space. When the Schrodinger equation is solved we obtain a general expression for .

Boundary conditions are then applied that are associated with the particular physical condition. For an electron in an atom, the boundary conditions are that ² must be continuous, single valued and finite everywhere. This makes sense as probability functions do not fluctuate radically from one place to another; the probability of finding the electron a few thousandths of a nanometer from a given location will not be radically different from the probability at the original position. Second the probability of finding an electron in a given place cannot have two different values simultaneously. Third, since the probability of finding an electron somewhere must be 100% or 1.000, if the electron really exists, the probability at any one point cannot be infinite.

According to quantum mechanics the electron is spread out like a wave, and because of Heisenberg's uncertainty principle we can only speak of the probability of finding the electron at some point in space. The wave which describes how the electron is distributed, spacially, is called a wave function. And like any other wave, the wave which describes an electron has regions where the value of the wave is large and regions where the value of the wave is small. These wave functions are also called atomic orbitals and can be thought of as regions of space where the probability of finding the electron is high. Recall that for the Bohr atom the region of space where the probability of finding the electron was defined as an orbit of a particular radius. This was a two dimensional description which was inadequate. In quantum mechanics we must describe the electron in three dimensional shapes. In the the Bohr model the principle quantum number, n, was all that was needed to calculate the energy and the location of the electron because all of the shapes of the orbits were identical..a circle. In quantum mechanics the atomic orbitals require three quantum numbers to complete describe the energy and the shape the electron occupies. In the Bohr model we could describe the elctron as in an orbit. But in the quantum mechanical model the electron occupies an orbital.

But since we are talking about additional quantum numbers we need to take a minute to discuss the relationship of these new quantum numbers. Remember a quantum number is a whole number which is used to label the of an electron in an atom.

Since an orbital is 3-dimensional, the solutions to the Schrodinger equation for the hydrogen atom are characterized by three integer quantum numbers: n, l, and m_l.

The three quantum numbers that are obtained from the wavefunction are referred to as 'n' the principal quantum number, 'l' the azimuthal quantum number and 'm_l' the magnetic quantum number.

The three quantum numbers n, l, and m_l are mathematically related to one another. The principal quantum number ('n') can have whole number values equal to 1, 2, 3, 4, .....

The principle quantum number (n) establishes the energy of the electron and the size of the orbital (distance from the nucleus). We also use the term shell in reference to 'n' value of an electron. The azimuthal quantum number ('l') can have values that are related to 'n'. 'l' can have integer values that range from 0 to n-1.

For example, if n=1, then 'l' can only have one value and it is l=0. If n=2, then 'l' can have two values, 0 and 1. The 'l' quantum numbers each have a distinctive shape which is characteristic of the value of 'l'. All the 'l' values associated with a particular 'n' value are called subshells. When n=1, 'l' can have only one value, therefore there is one subshell associated with the n=1 shell. If n=2 there are two subshells. The magnetic quantum number can have values which depend on 'l'. They are related to 'l' in the following way;

m_l = -'l', -('l'+1), ... 0 ..., +('l'-1), +'l'

The magnetic quantum number defines the number of orbitals in a subshell. In the l=0 subshell there is only one m_l value, therefore one orbital is associated with the l=0 subshell. If l=1 then there are three orbitals.

So the set of quantum numbers define the state of an electron. Taken together the three quantum number specify the orbital the electron occupies. They also specify the energy of the orbital, the shape of the orbital and the orientation of the orbital.

It becomes combersome to have to attach labels with three quantum numbers to each possible state for an electron. To simplify the notation a variation is used to label the orbital the electron can occupy. In this variation the numeric value of the principal quantum number is retained. However, the azimuthal quantum number is identified with a letter: s (l=0), p (l=1), d (l=2), f (l=3). We use these letters to denote the subshells and we speak of a 1s subshell (n=1,l=0) a 2s subshell (n=2,l=0) a 2p subshell (n=2,l=1) etc. The magnetic quantum number and the spin quantum number are combined. The m_l values are what we referred to as orbitals and each orbital can have a maximum of two electrons. There for an 's' subshell can only have one orbital (m_l=0) which contains a maximum of two electrons. A 'p' subshell can have three orbitals (m_l = -1,0,+1) each of which can hold a maximum of two electrons. Therefore the 'p' subshell has three orbitals which can hold, together, six electrons. The 'd' subshell has five orbitals for a total of ten electrons. We can place all of these orbitals on an energy diagram to show the orbital energies in the hydrogen atom.

Each of the orbitals for the quantum states differentiated by n, l, and m_l corresponds to a different probability distribution function for the electron in space. The simplest such probability functions, for 's' orbitals (l = 0), are spherically symmetrical. The probability of finding the electron is the same in all directions but varies with the distance from the nucleus. The electron is no longer in an Bohr orbit of fixed distance from the nucleus, rather, it is an electron probability cloud. These probability clouds are called hydrogenlike atomic orbitals. The 2s orbital is also spherically symmetrical, but its radial distribution function has a node, that is, zero probability, at r = .1058 nm from the nucleus. The highest density corresponds to the Bohr radius of .212 nm. There is also a high probability of finding the electron closer to and further from the nucleus than the most probable of .212 nm. The 3s orbital is larger in size than both the 1s and 2s orbitals. There are three 2p orbitals, 2p_x, 2p_y, and 2p_z. Each orbital is cylinderically symmetrical with respect to rotation around one of the three principal axes x, y, or z, as identified by the subscript. Each p orbital has two lobes of high probability density separated by a nodal plane of zero density. The sign of the wavefunction, &fnof, is positive in one lobe and negative in the other. The 3p, 4p and higher p orbitals have one, two or more additional nodal shells around the nucleus (details of secondary importance). The important feature is that the orbitals get larger as the principal quantum number increases are mudally perpendicular and strongly directional.

The five d orbitals first appear for n = 3. The d_xz, d_yz, and d_xy are identical in shapebut differ in orientation. Each has four lobes of electron density bisecting the angles between the principal axes. The remaining two are unusal: the d_x2_-y2 orbital has lobes of density along the x and y axes, and the d_z2 orbital has lobes along the z axes, with a small doughnut or ring in the xy plane.

So the azimuthal quantum number can also be called the orbital-shape quantum number: s orbitals are spherical, p orbitals are cylinderical extensions of two lobes along the principal axes and d orbitals have extensions along two mutually perpendicular directions. The third quantum number describes the orientation of the orbital in space. It is also called the magnetic quantum number because in a magnetic field orbitals with with different spatial orientations have different energies.

Here is an animation with all of this stuff put together.