Wavefunction of a particle

Quantum mechanics explains how entities like electrons have both particle-like and wave-like characteristics. The SchrOdinger equation describes the wavefunction of a particle ... 

Although there is a strong analogy between photons and materiaF particles such as electrons, this should not be pushed too far. The wave theory of radiation describes oscillating electric and magnetic fields that can in principle be measured directly. The wavefunction of a particle does not seem to correspond to any physical field that can be directly detected experimentally in this way. It should strictly be regarded simply as a mathematical device used for calculation. 

Elementary quantum mechanics showed that a plane wave exp (iK r) has the same dependence on space and time as the wavefunction of a particle with momentum hK. 

The simplest model describing general features of hydrogen tunneling is that of a particle in a double-well potential (Eig. 26.9). If the positions of the two minima are close to each other, and the barrier between them is not too high, one may expect an overlap of the ground-state wavefunctions of a particle oscillating in... 

We assume that wavefunctions of the conjugated electrons in the linear polyene can be approximated by the wavefunctions of a particle in a one-dimensional box. Then... 

Consider a three-dimensional hox of lengths a = 2.0 A, b = 4.0 A, and c = 1.0 A. Identify the quantum numbers n ny,n for the wavefunction of a particle in this box when the spacings between the nodes along the x axis, the y axis, and the z axis are all equal, and when there is exactly one node along the z axis. 

Fig. 9.36 The wavefunction of a particle on the surface of a sphere must satisfy two cyclic boundary conditions. The wavefunction must reproduce itself after the angles 0 and 0 are swept by 360° (or 2jt radians). This requirement leads to two quantum numbers for its state of angular momentum.

Assume that the electronic states of the it electrons of a conjugated molecule can be approximated by the wavefunctions of a particle in a one-dimensional box and that the dipole moment can be related to the displacement along this length by /(= -ex. Show that the transition probability for the transition n=l n = 2is nonzero, whereas that for n = 1 —5 n = 3 is zero. Hint The following relations will be useful ... 

Soon after the 1926 Schrodinger paper, Thomas and Fermi derived a kinetic energy formula based on Jellium. The derivation, which requires only the wavefunctions of a particle in a box (see Section 9), shows that the exact alpha kinetic energy of jellium is... 

Replacing the integrand according to the trigonometry formula sin fcc = (l/2)-(l/2) cos fcc, we can arrive at = 2/L. Finally, the expression for the wavefunction of a particle inside the rectangular infinitely deep potential box is given as... 

Figure 7.8 Wavefunctions of a particle in a one-dimensionai infiniteiy deep potential box (a) and...

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