Wavefunctions Born interpretation

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... 

Born interpretation The interpretation of the square of the wavefunction, i j, of a particle as the probability density for finding the particle in a region of space. 

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 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... 

For a single-particle system, the wavefunction T(r, t), or i/ (r) for the time-independent case, represents the amplitude of the still vaguely defined matter waves. The relationship between amplitude and intensity of electromagnetic waves developed for Eq (2.6) can be extended to matter waves. The most commonly accepted interpretation of the waveliinction is due to Max Born (1926), according... 

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). 

Readers who can remember what their first course in chemistry was like may recall how hard it was to learn how to visualize molecular structures and from there to visualize electronic wavefunctions. It is fair to say that an even more difficult challenge awaits those who attempt to visualize the phase space of reactive classical molecular dynamics within the Born-Oppenheimer approximation. The term phase space seems to have been originally coined by Gibbs. 2 But what is phase space And if it is so hard to visualize, why should we want to visualize it Obviously, such an endeavor is worthwhile to the extent that it yields new insights about molecular motion and helps us to interpret experiments. The concept of phase space will gradually emerge from the discourse in the remainder of this section. 

While the electron wavefunction can be used to obtain the energy and other properties of the electron, the question arises, in quantum mechanics generally, as to what the wavefunction itself means . This has been, and still is, the subject of much debate and there is currently intense research activity into using attosecond spectroscopy to probe atomic wavefimctions [16]. The most useful interpretation of the wavefunction for chemistry is that due to Born, who, by analogy to a light wave, where the intensity is proportional to the square of the amplimde, suggested... 

FIGURE 10.3 Max Born (1882-1970). His interpretation of the wavefunction as a probability rather than an actuality changed the common understanding of qiMntum mechanics. 

According to Max Born s interpretation of the wavefunction the square of the absolute value of the wavefunction of an electron, = V (a) gives the probability... 

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