Schrodingers Standing Waves

Schrodinger started out from classical mechanics in the way it was derived by the mathematician W. R. Hamilton (1805-1865). The Hamilton function in this theory is simply the sum of kinetic energy, expressed as a function of the momentum p and potential energy V, expressed as a function of the position of the particle. According to Schrodinger, physical observables must be replaced by operators. 

An operator transforms a function to another one. The operator may be a function that multiplies the function it acts on. For example, the operator x transforms the function sin(x) into the new function x sin(x). Alternatively, the action of the operator may be to perform a derivation. The derivative operator transforms the function sin(x) to the function cos(x). 

Other physical variables than kinetic and potential energy also correspond to an operator. The operator for momentum is a derivative multiplied by Planck s constant and (-i), where i is the imaginary unit. If we let that operator act on (onedimensional) matter waves, we obtain 

The momentum operator acting on the matter wave thus gives the product of the momentum and the matter wave. The kinetic energy p /2m may thus be replaced by a second derivative  

this operator always represents the kinetic energy of a single electron. 

---

By 1926, just in time for Davidson and Germer s 1927 experiment, Schrodinger put into mathematical form an idea due to de Broglie (1924). It was that the sometimes wavelike character of electrons could be the basis of the quantum states. The waves had to fit into the space available (e.g., the distance between two nuclei in a solid), and it was this need to fit and make a standing wave that made only certain states—certain wavelengths (or energies)—possible. 

The only purpose of the box is to allow normalization of the translational wave function, i.e. the exact size is not important. The solutions to the Schrodinger equation for such a particle in a box are standing waves, cosine and sine functions, and the energy levels are very close together. The summation in the partition function can therefore be replaced by an integral (an integral is just a sum in the limit of infinitely-... 

Unbeknownst to Schrodinger, there is a singularity imposed when the one-electron functions are superposed in a concentric manner. Should one treat the one-electron functions as standing waves with centers displaced from each other by an amount e, see Figure 1, and take the square magnitude of the resultant spectral interference as the physical quantity, the result as e -> 0 does not necessarily equal the result at e = 0. This had a very deleterious effect on his calculations. 

When the Schrodinger equation is solved for the hydrogen atom, it is found that only certain standing wave solutions exist. These solutions for one electron are known as orbitals, a term that derives from an earlier model of the atom in which electrons occupied orbits around the nucleus as the planets do around the Sun. If we work out the energies of these orbitals for the hydrogen atom, we find they correspond to the experimentally found energy levels n = 1, n = 2, n = 3, etc. The distribution in Figure 3.3 is that of an electron in the Is orbital (that is, one with n = 1 and l = 0). This electron has a spherical distribution in space it is equally likely to be found in any direction. However, its distribution varies with distance from the nucleus. The electron is much more likely to be close to the nucleus than far away. 

Standing wave solutions to the Schrodinger equation for an electron in an atom or molecule are known as orbitals. 

It is important to recognize that Schrodinger could not be sure that this idea would work. The test had to be whether or not the model would correctly fit the experimental data on hydrogen and other atoms. The physical principles for describing standing waves were well known in 1925 when Schrodinger decided to treat the electron in this way. Flis mathematical treatment is too complicated to be detailed here. Flowever, the form of Schrodinger s equation is... 

Pauli s research would lead to his receipt of the 1945 Nobel Prize in physics. In 1925, physicist Friedrich Hund (1896-1997) explained atomic spectroscopic data with a rule of maximum multiplicity Electrons are added to build up an atom so that the maximum number of energy levels (of equal energy) is filled with one electron each before electrons are paired. In 1926, Erwin Schrodinger (1887-1961), then at the University of Zurich, extended de Broglie s concept and treated electrons in atoms (and molecules) as standing waves and derived the new quantum mechanics. Electronic properties are determined by solving for the wave function, P, and energy for an atom or molecule. 

When the Hamilton function had been replaced by the Hamiltonian operator, SchrOdinger was able to write down a time-independent (Schrodinger) equation (SE) for standing waves as a function of the electron coordinates ... 

Earlier we saw that we needed a wave equation in order to solve for the standing waves pertaining to a particular classical system and its set of boundary conditions. The same need exists for a wave equation to solve for matter waves. Schrodinger obtained such an equation by taking the classical time-independent wave equation and substituting de Broglie s relation for A. Thus, if... 

The first of these equations is just the time-independent Schrodinger equation we have been using. The second equation has the solution fit) = Aexpi—iEt/h). Hence,/ / equals a constant, and so4 4 = / // /ai/r /. Since / has no effect on energy or particle distrihution, we can ignore it in dealing with stationary states. The situation is analogous to the case of standing waves discussed in Chapter 1. 

In the formal theory of quantum mechanics, the Schrodinger wave equation is taken as a postulate (fundamental hypothesis). In order to demonstrate a relationship with the classical wave equation, we obtain the time-independent Schrodinger equation nonrigorously for the case of a particle that moves parallel to the x axis. For a standing wave along the x axis, the classical coordinate wave equation of Eq. (14.3-10) is... 

We will see that a solution to the time-independent Schrodinger equation provides both a coordinate wave function if and an energy value E. We can immediately write a solution to the time-dependent equation by multiplying a coordinate wave function by the time factor. This type of solution, with the coordinate and time dependence in separate factors, corresponds to a standing wave, because any nodes are stationary. There are also solutions of the time-dependent Schrodinger equation that are not products of a coordinate factor and a time factor. These solutions can correspond to traveling waves. 

---