Born interpretation

The quantum mechanical state n) has no direct physical interpretation, but its absolute square, Y p=Y Y , can be interpreted as a probability density distribution. This so alled Born interpretation implies for a single particle that the wave function has to be normalized, i.e., integration over all dynamical variables of a system must yield unity. 

We define the particle density distribution of the quantum mechanical state Y as 

For the sake of brevity, we will usually drop the state index n and write simply p r) for the particle density distribution as it will be clear from the context whether this is a ground state density or the density of a state higher in energy. 

After integration of the density of any state over all space the number of particles is recovered. 

In the case of purely electronic systems, the particle density is also called electron density. Trivially related to the electron density is the charge density 

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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... 

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. 

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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%. 

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