Bom interpretation of the wavefunction

FIGURE 1.24 The Bom interpretation of the wavefunction. The probability density (the blue line) is given by the square of the wavefunction and depicted by the density of shading in the band beneath. Note that the probability density is zero at a node. A node is a point where the wavefunction (the orange line) passes through zero, not merely approaches zero. 

The function ij/(r, 9, p) (clearly ij/ could also be expressed in Cartesians), depends functionally on r, 6, p and parametrically on n, l and inm for each particular set (n. I, mm ) of these numbers there is a particular function with the spatial coordinates variables r, 6, p (or x, y, z). A function like /rsiiir is a function of x and depends only parametrically on k. This ij/ function is an orbital ( quasi-orbit the term was invented by Mulliken, Section 4.3.4), and you are doubtless familiar with plots of its variation with the spatial coordinates. Plots of the variation of ij/2 with spatial coordinates indicate variation of the electron density (recall the Bom interpretation of the wavefunction) in space due to an electron with quantum numbers n, l and inm. We can think of an orbital as a region of space occupied by an electron with a particular set of quantum numbers, or as a mathematical function ij/ describing the energy and the shape of the spatial domain of an electron. For an atom or molecule with more than one electron, the assignment of electrons to orbitals is an (albeit very useful) approximation, since orbitals follow from solution of the Schrodinger equation for a hydrogen atom. 

According to the Bom interpretation of the wavefunction, lelectron density of electron 1 in orbital IcTg at a position ri. Similarly, lag(r2)lcTg(r2) is the electron density of electron 2. The electrostatic repulsion between these regions of electron density thus equals lcTg(ri)lag(ri) x (l/ri2) x lag(t2)l(Tg(t2), where rx2 is the distance between the two electrons. The integral of this function over aU space thus corresponds to the electrostatic (Coulomb) repulsion between the two orbitals. 

The electron density p(r) at a point r can be calculated from the Bom interpretation of the wavefunction as a sum of squares of the spin orbitals at the point r for all occupied molecular orbitals. For a system of N electrons occupying N/2 real orbitals, we can write ... 

The uncertainty principle shows that the classical trajectory of a particle, with a precisely determined position and momentum, is really an illusion. It is a very good approximation, however, for macroscopic bodies. Consider a particle with mass I Xg, and position known to an accuracy of 1 pm. Equation 2.41 shows that the uncertainty in momentum is at least 5 x 10 29 kg m s-1, corresponding to a velocity of 5 x 10 JO m s l. This is totally negligible for any practical purpose, and it illustrates that in the macroscopic world, even with very light objects, the uncertainty principle is irrelevant. If we wanted to, we could describe these objects by wave packets and use the quantum theory, but classical mechanics gives essentially the same answer, and is much easier. At the atomic and molecular level, however, especially with electrons, which are very light, we must abandon the idea of a classical trajectory. The statistical predictions provided by Bom s interpretation of the wavefunction are the best that can be obtained. 

FIGURE 12.12 Max Bom (1882-1970). Not only did he develop the probabilistic interpretation of the wavefunction, but he also devised a quantum-mechanical description for molecules. 

Wavefunctions by themselves can be very beautiful objects, but they do not have any particular physical interpretation. Of more importance is the Bom interpretation of quantum mechanics, which relates the square of a wavefunction to the probability of finding a particle (in this case a particle of reduced mass /r vibrating about the centre of mass) in a certain differential region of space. This probability is given by the square of the wavefunction times dx and so we should concentrate on the square of the wavefunction rather than on the wavefunction itself. 

Even worse is the confusion regarding the wavefunction itself. The Bom interpretation of quantum mechanics tells us that T/r (r) (r)dr represents the probability of finding the particle with spatial coordinates r, described by the wavefunction i//(r), in volume element dr. Probabilities are real numbers, and so the dimensions of jr(r) must be of (length)-3 2. In the atomic system of units, we take the unit of wavefunction to be ao 3/2-... 

The interpretation of the square of the wavefunction as a measure of electron density in atoms and molecules arose from a slightly different suggestion by Max Bom Bom M (1926) Z Phys 37 863. See Moore W (1989) Schrodinger. Life and thought. Cambridge University Press, Cambridge, UK, pp 219-220, 225-226, 240, 436-436... 

Max Born, German—British physicist. Bom in Breslau (now Wroclaw, Poland), 1882, died in Gottingen, 1970. Professor Berlin, Cambridge, Edinhurgh. Nohel prize, 1954. One of the founders of quantum mechanics, originator of the prohahility interpretation of the (square of the) wavefunction (chapter 4). 

In the Bom interpretation (section 4.2.6) the square of a one-electron wavefunction at any point X is the probability density (with units of volume" ) for the wavefunction at that point, and dx dy dz is the probability (a pure number) at any moment of finding the electron in an infinitesimal volume dxdydz around the point (the probability of finding the electron at a mathematical point is zero). For a multielectron wavefunction 4 the relationship between thewavefunction 4 and the electron density p is more complicated (involving the summation over all spin states of all electrons of -fold integrals of the square of the wavefunction), but it can be shovra [7] that piyc, y, z) is related to the component one-electron spatial wavefunctions of a single-determinant wave-... 

We know from everyday experience what is meant by the classical state of a particle—that is, by position and momentum. But what is meant by i/r The currently accepted physical interpretation of f, given in 1926 by German physicist Max Bom, is that the wavefunction is related to the probability of finding the particle in a specific region of space. Because ip can take on negative values, and probability is, by definition, a positive quantity. Bom po ifialM Jhj the - lifitjMfj in... 

The physical connection of the wavefunction, vp. must still be determined. The basis for the interpretation of v / comes from a suggestion made by Max Bom in 1926 that v corresponds to the square root of the probability density the square root of the probability of finding a particle per unit volume. The wavefunction, however, may be a complex function. As an example for a given state n. 

The wavefunction for Region II contains an exponentially increasing component as x increases. As x approaches infinity, the wavefunction in Region II will approach infinity. This is an untenable result based on the Bom interpretation - the probability density of the particle will approach infinity as wavefunction approaches infinity. Since this is not physically possible, the positive exponential component of the wavefunction must be discarded. 

Then a mathematical statement of Bom s interpretation is iff2 dx is equal to the probability of finding the particle in the region between x and x + dx. A wavefunction which satisfies eqn 2.34 is said to be normalized. This condition is necessary for the strict statement of Bom s interpretation, as the... 

Consider first the situation where a particle moves in two dimensions, labelled x and y. The wavefunction now depends on these two variables. As before, the Bom interpretation shows that its square gives the probability of finding the particle at some position (x,y). The two-dimensional form of Schrodingef s equation is... 

The Bom interpretation leads to a number of important implications on the wavefunction. The function must be single-valued it would not make physical sense that the particle had two different probabilities in the same region of space. The sum of the probabilities of finding a particle within each segment of space in the universe ( /n Vn times a volume element, dx) must be equal to unity. The mathematical operation of ensuring that the sum overall space results in unity is referred to as normalizing the wavefunction. 

In order to understand the spectroscopic signatures of water, we shortly intfoduce some concepts how molecular spectra can be understood and interpreted. In the Bom-Oppenheimer Approximation it is assumed that the motions of the electrons and the nuclei of a molecule can be separated. If r, denotes the positions of the electrons i and Ry the positions of the nuclei j, then the wavefunctions are given by... 

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