The Born Interpretation

So far a model has been developed to obtain the energy of the system (an experimentally determinable property - i.e. an observable) by applying an operator, the Hamiltonian, to the wavefunction for the system. This approach is analogous to how the energy of a classical standing wave is obtained. The second derivative with respect to position is taken of the function describing the classical standing wave. 

The major difference between the quantum mechanical approach for describing particles and that of classical mechanics describing standing waves is that in classical mechanics the operator (taking the second derivative with respect to position) is applied to a function that is physically observable. At this point, the wavefimction describing the particle has no observable property beyond the de Broglie wavelength. 

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 square of this function will result in a complex value. To ensure that the probability density has a real value, the probability density is obtained by multiplying the wavefunction by the complex conjugate of the wavefunction. The complex conjugate is obtained by replacing any i in the function with a -i . The complex conjugate of the function above is 

Consider a 1-dimensional system where a particle is free to be found anywhere on a line in the x-axis. Divide the line into infinitesimal segments of length dx. The probability that the particle is between x and x -t dx is / /ndx, It is important to note that ln Wn is not a probability but rather it is a probability density (i.e. probability per unit volume). To find the probability, the product must be multiplied by a volume element (in the case of a 1-dimensional system, the volume element is just dx). 

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

The Born interpretation of quantum mechanics tells us that s)dTds gives the chance of finding the electron in the spatial volume element dr and with spin coordinate between s and s + ds. Since probabilities have to sum to 1, we have... 

The Born interpretation of as a probability function requires that the wavefunction ei be normalized, namely that integration of dv over all space (Equation 1.11), equals... 

Wavefunctions must be either symmetric (delete the minus sign from Equation 1.12) or antisymmetric in order to be consistent with the Born interpretation electrons being indistinguishable, W2 must be invariant with respect to an interchange of any pair of electrons, because the probability of finding e, in a volume element around the coordinates qej and ey around qe. must be the same when the labels / and j are exchanged. Both symmetric and antisymmetric wavefunctions would satisfy this condition, but the Pauli principle allows only antisymmetric wavefunctions. 

The Born interpretation also requires that wavefunctions be either symmetric or antisymmetric with respect to all symmetry operations of a molecule, that is, when the coordinates of all the electrons and nuclei are exchanged by symmetry-equivalent coordinates. For example, the electronic distribution around an isolated atom must be spherically symmetric in the absence of external fields. 

The acceptable solutions to the one-dimensional particle in a box problem are sketched in Figure 3.27(a) for the first several quantum numbers. The Born interpretation of the wave function states that the product y/ i// represents the probability density of finding the electron in a finite region of space. Because the Born interpretation of the wave function is this function is shown in Figure 3.27(b). 

Born, Max (1882-1970) German physicist who was one of the founders of quantum mechanics in the 1920s. In particular, he put forward the Born interpretation for the wave-function of an electron in terms of probability in 1926. Born also made major contributions to the theory of crystals and to the quantum theory of molecules. He was awarded a share of the 1954 Nobel Prize in physics (together with Walther Bothe) for his work on quantum mechanics. 

The Born interpretation affects the entire meaning of quantum mechanics. Instead of giving the exact location of an electron, it will provide only the probability of the location of an electron. For those who were content with understanding that they could calculate exactly where matter was in terms of Newton s laws, this interpretation was a problem because it denied them the ability to state exactly how matter was behaving. All they could do was state the probability that matter was behaving that way. Ultimately, the Born interpretation was accepted as the proper way to consider wavefunctions. 

Using the Born interpretation, for an electron having a one-dimensional wavefunction " F = sin 7TX in the range x = 0 to 1, what are these probabilities ... 

The Born interpretation makes obvious the necessity of wavefunctions being bounded and single-valued. If a wavefunction is not bounded, it approaches infinity. Then the integral over that space, the probability, is infinite. Probabilities cannot be infinite. Because probability of existence represents a physical observable, it must have a specific value therefore, P s (and their squares) must be single-valued. 

Because the wavefunction in this last example does not depend on time, its probability distribution also does not depend on time. This is the definition of a stationary state A state whose probability distribution, related to P(x)p by the Born interpretation, does not vary with time. 

The Born interpretation suggests that there should be another requirement for acceptable wavefunctions. If the probability for a particle having wavefunction were evaluated over the entire space in which the particle exists, then the probability should be equal to 1, or 100%. In order for this to be the case, wavefunctions are expected to be normalized. In mathematical terms, a wavefunction is normalized if and only if... 

The integral s limits would be modified to represent the limits of the space a particle inhabits (we will see examples shortly). What equation 10.8 usually means is that wavefunctions must be multiplied by some constant, called the normalization constant, so that the area under the curve of is equal to 1. According to the Born interpretation of normalization also guarantees that the probability of a particle existing in all space is 100%. 

Before going any further, it will be helpful to understand the physical significance of a wavefunction. The interpretation most widely used is based on a suggestion made by the German physicist Max Born. He made use of an analogy with the wave theory of light, in which the square of the amplitude of an electromagnetic wave is interpreted as its intensity and therefore (in quantum terms) as the number of photons present. The Born interpretation asserts ... 

The amplitude of a Is orbital depends only on the radius, r, of the point of interest and is independent of angle (the latitude and longitude of the point). Therefore, the orbital has the same amplitude at all points at the same distance from the nucleus regardless of direction. Because, according to the Born interpretation (Section 9.2b), the probability density of the electron is proportional to the square of the wavefunction, we now know that the electron will be found with the same probability in any direction (for a given distance from the nucleus). We summarize this angular independence by saying that a Is orbital is spherically symmetrical. Because the same factor Y occurs in all orbitals with / = 0, all s orbitals have the same spherical symmetry (but different radial dependences). 

We often need to know the total probability that an electron will be found in the range r to r -l- 6r from a nucleus regardless of its angular position (Fig. 9.46). We can calculate this probability by combining the wavefunction in eqn 9.33 with the Born interpretation and find that for s orbitals, the answer can be expressed as... 

The experiments that generate charge density maps show us how the electrons as a whole are distributed in a molecule, but do not show the orbitals they occupy. In the Born interpretation, the electron density p(r) at a point r is linked to the set of occupied MOs ( i(r) by a summation of the wavefunction magnitudes squared ... 

For the corresponding electron density we can make use of the Born interpretation of the wavefunction again. This states that the product of the wavefunction and its own complex conjugate gives the probability per unit volume of finding an electron occupying the orbital... 

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