Probability density defining wavefunction

The analysis of this wavefunction in terms of contour maps of a weighted scattering wavefunction probability density defined by... 

For example, if motion is constrained to take place within a rectangular region defined by 0 < x < L 0 < y < Ly, then the continuity property that all wavefunctions must obey (because of their interpretation as probability densities, which must be continuous) causes A(x) to vanish at 0 and at Lx. Likewise, B(y) must vanish at 0 and at Ly. To implement these constraints for A(x), one must linearly combine the above two solutions exp(ix(2mEx/h2)1 /2) and exp(-ix(2mEx/h2)l/2) to achieve a function that vanishes at x=0 ... 

From the Uncertainty Principle, we no longer speak of the exact position of an electron. Instead, the electron position is defined by a probability density function. If this function is called p (x,y,z), then the electron is most likely found in the region where p has the greatest value. In fact, p dr is the probability of finding the electron in the volume element dr (= dxdydz) surrounding the point (x,y,z). Note that p has the unit of volume-1, and pdr, being a probability, is dimensionless. If we call the electronic wavefunction f, Born asserted that the probability density function p is simply the absolute square of tjr. 

As mentioned earlier, density functional theory (DFT) does not yield the wave-function directly. Instead it first determines the probability density p and calculates the energy of the system in terms of p. Why is it called density functional theory and what is a functional anyway We can define functional by means of an example. The variational integral E(trial wavefunction ip and it yields a number (with energy unit) for a given electron density p, which itself is a function of electronic coordinates. 

While the trapped electron could be viewed as the simplest possible anion, there is a significant difference between alkalides and electrides. Whereas the large alkali metal anions are confined to the cavities, only the probability density of a trapped electron can be defined. The electronic wavefunction can extend into all regions of space, and electron density tends to seek out the void spaces provided by the cavities and by intercavity channels. 

We may graphically examine the wavefunctions and the probability densities as we vary the quantum number, n. There is no reason to expect that the wavefunctions will be the same. In fact, the solutions look as shown in Fig. 7.2. A node is defined as a point at which ip and ip 2 are zero. We will not find the particle at this position. The wavefunctions, ip are identical to the standing waves generated by the vibrating string, an example with which we are all much more familiar and has been treated in the previous chapter. Note especially, that the n = 1 wavefunction for the particle-in-a-box is identical to the string plucked at its midpoint in Fig. 6.1. 

Figure 2.3 Contour plots of several low-lying H atom orbitals. Curves are surfaces on which the wavefunction exhibits constant values solid and dashed curves correspond to positive and negative values, respectively. The outermost contour in all cases defines a surface containing 90% of the electron probability density. The incremental change in wavefunction value between adjacent contours is 0.04, 0.008, 0.015, 0.003, 0.005, and 0.003 bohr respectively for the Is, 2s, 2p, 3p, and 3d orbitals. Boxes exhibit side lengths of 20 bohrs (Is, 2s, 2p) and 40 bohrs (3s, 3p, 3c0, so that orbital sizes can be compared. Straight dashed lines in 3p and 3d 2 plots show locations of nodes.

If we define the wavefunction in terms of the probability density, and it s the probability density that determines what we measure in the laboratory, then why do we need the wavefunction at all The wavefunction is a mathematical convenience that allows us not only to predict the properties of individual quantum states, but also to predict results that stem from combinations of quantum states. It is the wavefunction, not the probability density, that solves the Schrodinger equation, and—like a wave—the wavefunction can undergo constructive and destructive interference with other wavefunctions to accurately predict the resulting probability densities. 

The real part of the term is equal to cosw/0, and it would contribute w/ additional nodal planes. Consequently, the total number of angular nodes in the real part of if/ is equal to /. Each angular node corresponds to a surface (a plane for nodes in (f), a cone for nodes in 6) on which the electron wavefunction vanishes. For example, the do wavefunction has nodes at 6 = 54.7° and 125.3°, and all the points at each value of 6 form a cone centered on the z axis. On the other hand, the real part of the pi wavefunction has a node at 0 = 90°, which defines the yz plane. The nodal planes along cf) vanish when Yf is multiplied by its complex conjugate, and therefore the probability density has no nodes along 4>, but retains the /— m/ ... 

An important distinction arises here between Cartesian and spherical coordinates the probability density can be inconsistently defined if we re not cautious. The probability density of a wavefunction R r) is k(r), but this gives the probability per unit volume of the particle being at some particular value of r. Often, however, we are not interested in any specific direction from the nucleus, and we want to know the density at some value of r added up over all the angles. In that case, the function we want is the radial probability density, equal to R i rYr because the volume sampled by the radial wave-function increases, like the surface area of a sphere, as r. Looking at it another way, when we integrate over the probability density, we add a factor of in the volume element. That is the same factor we include when we write the radial probability density. As shown in Fig. 3.11, the nodes that appear in... 

In the third area a motion once again is infinite and, accordingly, wavefunction is presented by periodic function and constant probability density. Its absolute magnitude is defined by one of the boundary conditions (condition of continuity). It Unks Am with Aj by the correlation... 

STM detects elecflic currents due to tunnel electrons between the sample and the probe tip. Tunneling probability at the tip position is dependent on the overlap of electronic wavefiinctions between the sample and the tip. Because the wavefunction of the electron at the tip is localized on a single atom, STM visualizes the electronic local density-of-states (LDOS) of the sample at tip position T and energy E with atomic resolution [50,51]. Operation principles of a near-field optical microscope is similar to that of an STM [52,53]. Instead of using tunnel electrons as in an STM, a near-field optical microscope uses tunnel photons between the sample and the near-field probe tip and visualizes photonic LDOS at position V" and frequency co. In general, LDOS is defined by the following equation [54]. 

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