The Schrodinger wave equation

1-8 THE SCHRODINGER WAVE EQUATION In 1926, the Austrian physicist Erwin Schrodinger presented the equation relating the energy of a system to the wave motion. The Schrodinger equation is commonly written in the form 

The Schrodinger equation is a complicated differential equation and is capable of exact solution only for very simple systems. Fortunately, one of these systems is the hydrogen atom. 

The solution of the Schrodinger equation for the hydrogen atom yields wave functions of the general form 

The basis of wave mechanics is the wave equation of Schrodinger which can be obtained by the combination of the de Broglie relationship 

Substitution of this expression for A in ti then gives the Schrodinger wave equation 

The steps leading to equation 1.34 must not be regarded as a rigorous derivation of the Schrodinger equation, since we have employed an equation of classical mechanics to obtain a wave mechanical expression from wliich the quantum condition of discrete energy levels may be devised. 

This illustration may be regarded as an example of Heisenberg s Uncertainty Principle, which can be stated in the following form The more accurately the velocity of an electron is defined, the less certainly is its position known and, conversely, the more accurate the definition of the position of an electron, the less precise is the value of its velocity Expressed mathematically this becomes 

This principle enables us to make an appreciation of the relationship which exists between wave mechanics and classical mechanics. The right hand side of equation 1.37 is inversely proportional to m and hence for a very heavy particle h/m is very small, which makes it possible for the product AxAv also to be very small. Under such circumstances it is therefore possible that both Ax and Av may be small and hence both the velocity and the position of the particle may be known with certainty. This applies in classical mechanics. When, however, m is very small as for atomic particles, A/m is large and thus either Ax or Av but not both may be small and hence the position and the velocity cannot at the same time both be known with certainty. This is true for the mechanics of very small particles i.e, wave mechanics. 

Information about the wavefunction is obtained from the Schrodinger wave equation, which can be set up and solved either exactly or approximately the Schrodinger equation can be solved exactly only for a species containing a nucleus and only one electron (e.g. H, He ), i.e. a hydrogen-like system. 

A hydrogen-like atom or ion contains a nucleus and only one electron. 

An important physical observation which is a consequence of the de Broglie relationship is that electrons accelerated 

The Schrddinger wave equation may be represented in several forms and in Box 1.3 we examine its application to the motion of a particle in a one-dimensional box equation 1.12 gives the form of the Schrodinger wave equation that is appropriate for motion in the x direction  

Of course, in reality, electrons move in three-dimensional space and an appropriate form of the Schrddinger wave equation is given in equation 1.13. 

Information about the wavefunction is obtained from the Schrodinger wave equation, which can be set up and 

Solving this equation will not concern us, although it is useful to note that it is advantageous to work in spherical polar coordinates (Fig. 1.4). When we look at the results obtained from the Schrddinger wave equation, we talk in terms of the radial and angular parts of the wavefunction, and this is represented in eq. 1.14 where R r) and A 6, / ) are radial and angular wavefunctions respectively.  

Having solved the wave equation, what are the results  

The time-dependent differential equation for the wave function W(x,t) bears the name of Schrodinger. For a one-dimensional case it is given by 

Equation (2.22) is a general equation that describes peculiarities of the objects in the microscopic world. The differential equation has particular solutions. 

It is easily seen that these formulas describe wave motion which we are already acquainted with. The wave function is a complex value. Let us check the solution (2.23) for V(x, t) = Vo = const. 

It might seem that the Schrodinger equation is unfounded. As a matter of fact, it cannot be derived either from laws of classic physic or from any known dependences. However, the equation is substantiated by correctly predicting of a number of natural phenomena. 

The wave function is linear in x, t). That is, if W x, t) and i(x, t) are two different solutions to the Schrodinger wave equation then any of their linear combinations. 

In his doctoral dissertation de Broglie postulated that particles such as the electron, proton, etc. should also possess wave-like properties in exact analogy with the particle-like properties exhibited by electromagnetic waves in the quantum theory of radiation. For motion in one dimension he postulated that the momentum of the particle p and its kinetic energy E were related to the wavevector k and angular frequency w of the guiding wave, I, by the relations 

In the case of a free particle of mass m moving along the x-axis, the energy and momentum satisfy the equation 

This suggests that the energy and momentum of a particle can be represented by differential operators which act on the wavefunction V(r,t) where 

For a particle of mass m moving in a potential V(iJ the sum of the kinetic and potential energy of the system is constant. SchrSdinger postulated, therefore, that equation (3.3) could be generalized to include this case and that the equation governing the motion of the particle is then 

The validity of Schrodinger s wave equation rests not on the plausibility arguments we have used to justify it but on the 

This chapter is primarily about wave mechanics since that is the most convenient way to introduce undergraduates to quantum mechanics using calculus. An equivalent form called matrix mechanics will be discussed briefly in a later chapter. Consider again the 1923 paper by De Broglie and the experiments that validated the particle-wave duality in Chapter 10. One might well ask that if there really is some wave that describes the behavior of particles, then is there an equation that the wave obeys Even today it is difficult to say what the wave is, but it may help to find an equation it obeys. Step back a moment to some basic calculus  

Perhaps you did not notice the pattern before but we can put this into a general form as 

We will now present a derivation of the Schrodinger equation that may not be the way he thought of it but that follows from limited use of calculus and simple algebra. Since De Broglie implied there is some sort of invisible, untouchable, mathematical pilot wave accompanying the motion of particles, we assume the general form of such a wave and relate it to its own second derivative. We will use i j since it is universally used for the wave function. 

This has major implications in that there is a mathematical (calculus) operator Hop, which represents the total energy of a particle (in only one dimension so far) and a function v[ , which incorporates the De Broglie condition and is an eigenfunction of the total energy operator. Note the left side of the equation must be in energy units since the right side is in terms of tot- 

We can carry the analysis of the energy units further. Classically (sophomore physics), the 

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In 1926 Erwin Schrodinger (1887-1961), an Austrian physicist, made a major contribution to quantum mechanics. He wrote down a rather complex differential equation to express the wave properties of an electron in an atom. This equation can be solved, at least in principle, to find the amplitude (height) of the electron wave at various points in space. The quantity ip (psi) is known as the wave function. Although we will not use the Schrodinger wave equation in any calculations, you should realize that much of our discussion of electronic structure is based on solutions to that equation for the electron in the hydrogen atom. 

The quantum number ms was introduced to make theory consistent with experiment. In that sense, it differs from the first three quantum numbers, which came from the solution to the Schrodinger wave equation for the hydrogen atom. This quantum number is not related to n, , or mi. It can have either of two possible values ... 

After the discovery of quantum mechanics in 1925 it became evident that the quantum mechanical equations constitute a reliable basis for the theory of molecular structure. It also soon became evident that these equations, such as the Schrodinger wave equation, cannot be solved rigorously for any but the simplest molecules. The development of the theory of molecular structure and the nature of the chemical bond during the past twenty-five years has been in considerable part empirical — based upon the facts of chemistry — but with the interpretation of these facts greatly influenced by quantum mechanical principles and concepts. 

The physical interpretation of the quantum mechanics and its generalization to include aperiodic phenomena have been the subject of papers by Dirac, Jordan, Heisenberg, and other authors. For our purpose, the calculation of the properties of molecules in stationary states and particularly in the normal state, the consideration of the Schrodinger wave equation alone suffices, and it will not be necessary to discuss the extended theory. 

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

The coefficients of the various p orbitals of complex molecules containing n systems are obtained by computer programming based on approximate solutions of the Schrodinger wave equation. 

Erwin Schrodinger developed an equation to describe the electron in the hydrogen atom as having both wavelike and particle-like behaviour. Solution of the Schrodinger wave equation by application of the so-called quantum mechanics or wave mechanics shows that electronic energy levels within atoms are quantised that is, only certain specific electronic energy levels are allowed. 

Solving the Schrodinger wave equation yields a series of mathematical functions called wavefunctions, represented by R (Greek letter psi), and their corresponding energies. 

Atomic orbitals are actually graphical representations for mathematical solutions to the Schrodinger wave equation. The equation provides not one, but a series of solutions termed wave functions t[ . The square of the wave function, is proportional to the electron density and thus provides us with the probability of finding an electron within a given space. Calculations have allowed us to appreciate the shape of atomic orbitals for the simplest atom, i.e. hydrogen, and we make the assumption that these shapes also apply for the heavier atoms, like carbon. 

Substitution of the potential energy for this harmonic oscillator into the Schrodinger wave equation gives the allowed vibrational energy levels, which are quantified and have energies Ev given by... 

The Schrodinger wave equation for a particle in the potential energy regions I and [ can be written as... 

A rigorous quantum-mechanical treatment of directed valence bonds has not been given, for the reason that the Schrodinger wave equation has not been rigorously solved for any complicated molecule. Several approximate treatments have, however, been carried out, leading in a... 

Pauling showed that the quantum mechanical wave functions for s and p atomic orbitals derived from the Schrodinger wave equation (Section 5.7) can be mathematically combined to form a new set of equivalent wave functions called hybrid atomic orbitals. When one s orbital combines with three p orbitals, as occurs in an excited-state carbon atom, four equivalent hybrid orbitals, called sp3 hybrids, result. (The superscript 3 in the name sp3 tells how many p atomic orbitals are combined to construct the hybrid orbitals, not how many electrons occupy each orbital.)... 

Unlike molecular mechanics, the quantum mechanical approach to molecular modelling does not require the use of parameters similar to those used in molecular mechanics. It is based on the realization that electrons and all material particles exhibit wavelike properties. This allows the well defined, parameter free, mathematics of wave motions to be applied to electrons, atomic and molecular structure. The basis of these calculations is the Schrodinger wave equation, which in its simplest form may be stated as ... 

I think that the theory of resonance is independent of the valence-bond method of approximate solution of the Schrodinger wave equation for molecules. I think that it was an accident in the development of the sciences of physics and chemistry that resonance theory was not completely formulated before quantum mechanics. It was, of course, partially formulated before quantum mechanics was discovered and the aspects of resonance theory that were introduced after quantum mechanics, and as a result of quantum mechanical argument, might well have been induced from chemical facts a number of years earlier. [25]... 

Is resonance a real phenomenon The answer is quite definitely no. We cannot say that the molecule has either one or the other structure or even that it oscillates between them. .. Putting it in mathematical terms, there is just one full, complete and proper solution of the Schrodinger wave equation which describes the motion of the electrons. Resonance is merely a way of dissecting this solution or, indeed, since the full solution is too complicated to work out in detail, resonance is one way - and then not the only way - of describing the approximate solution. It is a calculus , if by calculus we mean a method of calculation but it has no physical reality. It has grown up because chemists have become used to the idea of localized electron pair bonds that they are loath to abandon it, and prefer to speak of a superposition of definite structures, each of which contains familiar single or double bonds and can be easily visualizable. [30]... 

The position and energy of each electron surrounding the nucleus of an atom are described by a wave function, which represents a solution to the Schrodinger wave equation. These wave functions express the spatial distribution of electron density about the nucleus, and are thus related to the probability of finding the electron at a particular point at an instant of time. The wave function for each electron, F(r,6,), may be written as the product of four separate functions, three of which depend on the polar coordinates of the electron... 

Surfaces may be drawn to enclose the amplitude of the angular wave function. These boundary surfaces are the atomic orbitals, and lobes of each orbital have either positive or negative signs resulting as mathematical solutions to the Schrodinger wave equation. 

The quantum number, m , originating from the 0(6) and Schrodinger wave equation, indicates how the orbital angular momentum is oriented relative to some fixed direction, particularly in a magnetic field. Thus, ml roughly characterizes the directions of maximum extension of the electron... 

Orbitals. Atomic orbitals represent the angular distribution of electron density about a nucleus. The shapes and energies of these amplitude probability functions are obtained as solutions to the Schrodinger wave equation. Corresponding to a given principal quantum number, for example n = 3, there are one 3s, three 3p and five 3d orbitals. The s orbitals are spherical, the p orbitals are dumb-bell shaped and the d orbitals crossed dumb-bell shaped. Each orbital can accomodate two electrons spinning in opposite directions, so that the d orbitals may contain up to ten electrons. 

Although the Schrodinger wave equation is difficult to solve for increasingly complicated atoms and molecules, we could, if we had a large enough computer, deduce the properties of all known chemicals from this equation. Quantum mechanics, however, is even more powerful than an ordinary cookbook because it also allows us to calculate the properties of chemicals that we have yet to see in nature. 

Molecular mechanics simulations are useful methods when dealing with large molecules or when limited information is required. When more sophisticated analysis is desired, such as thermodynamic data, it is usually necessary to switch to ab initio methods that seek to solve, or approximate a solution to, the Schrodinger wave equation for the entire molecule. Programs are rapidly improving both in terms of time taken to generate a solution to the molecular orbital and in the size of molecule that can be analyzed by these methods. Despite these advances only the simplest of supramolecular systems can usefully be investigated at this level of... 

The Schrodinger wave equation imposes certain mathematical restrictions on the quantum numbers. They are as follows ... 

In 1926, Erwin Schrodinger made use of the wave character of the electron and adapted a previously known equation for three-dimensional waves to the hydrogen atom problem. The result is known as the Schrodinger wave equation for the hydrogen atom, which can be written as... 

Predicting the optical spectrum requires the solution of the Schrodinger wave equation in terms of the states ( a) and energy levels (Ea) of the system and the Hamiltonian... 

The Schrodinger wave equation, Hip = Eip, lies at the heart of the quantum mechanical description of atoms. Recall from the preceding discussion that H represents an operator (the Hamiltonian) that extracts the total energy E (the sum of the potential and kinetic energies) from the wave function. The wave function ip depends on the x, y, and z coordinates of the electron s position in space. 

TABLE 12.1 Solutions of the Schrodinger Wave Equation for a One-Electron Atom n ( m( Orbital Solution... 

A quantum-mechaiucal calculation commences with the Schrodinger wave equation ... 

While a great deal of progress has proved possible for the case of the hydrogen atom by direct solution of the Schrodinger wave equation, some of which will be summarized below, at the time of writing the treatment of many-electron atoms necessitates a simpler approach. This is afforded by the semi-classical Thomas-Fermi theory [4-6], the first explicit form of what today is termed density functional theory [7,8]. We shall summarize below the work of Hill et al. [9], who solved the Thomas-Fermi (TF) equation for heavy positive ions in the limit of extremely strong magnetic fields. This will lead naturally into the formulation of relativistic Thomas-Fermi (TF) theory [10] and to a discussion of the role of the virial in this approximate theory [11]. 

The Schrodinger wave equation In 1926, Austrian physicist Erwin Schrbdinger (1887-1961) furthered the wave-particle theory proposed by de Broglie. Schrbdinger derived an equation that treated the hydrogen atom s electron as a wave. Remarkably, Schrbdinger s new model for the hydrogen atom seemed to apply equally well to atoms of other elements—an area in which Bohr s model failed. The atomic model in which electrons are treated as waves is called the wave mechanical model of the atom or, more commonly, the quantum mechanical model of the atom. Like Bohr s model,... 

The modem view of atoms holds that although atoms are composed of three types of subatomic particles—protons, neutrons, and electrons—the chemical behavior of atoms is governed by the behavior of the electrons. Furthermore, the electrons do not behave according to the traditional concept of particles, but rather exhibit the characteristics of waves. In a single atom these waves can be described by the Schrodinger wave equation ... 

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