Physical Chemistry , 1st ed.

tions in terms that accepted the uncertainty principle, and the Born interpre- tationis generally considered to be the correct w ...

has a known solution. It is  a b sin^2 xdx 2 x  4 1 sin 2 xba where we have substituted into the general form of the integra ...

the space a particle inhabits. What equation 10.8 usually means is that wave- functions must be multiplied by some constant so t ...

The wavefunctionin the above example has not changed. It is still a sine function. However, it is now multiplied by a constant s ...

specific formula from classical mechanics. In terms of linear momentum px, kinetic energy is given by K 2 p m x 2 Schrödinger, ...

Example 10.9 Consider an electron confined to some finite system. The state of the electron is described by the wavefunction  ...

10.8 An Analytic Solution: The Particle-in-a-Box Very few systems have analytic solutions (that is, solutions that have a specif ...

what the eigenvalues for the energy are, because with identically zero, by the Born interpretation the particle has a zero prob ...

 2 m  2   x 2 2 (Acos kxBsin kx)  2 m  2   x (kAcos kxkBsin kx)   2 m ^2 (k^2 Acos kxk^2 Bsin kx) Fa ...

were, then would be zero between 0 and a, and then it would be zero every- where and the particle would not exist anywhere. We ...

to indicate this dependence. The acceptable wavefunctions for a one-dimensional particle-in-a-box are written as n(x)  ...

10.9 Average Values and Other Properties There are other common observables in addition to energy. One could oper- ate on the wa ...

Example 10.12 Determine x , the average value of the position of an electron having the lowest energy level (n1) in a particle- ...

Other properties can also be determined from for the particle-in-a-box. We point them out because they are properties that can ...

Example 10.13 -Carotenes are highly conjugated polyenes found in many vegetables. They can be oxidized and used to synthesize p ...

This system is illustrated in Figure 10.8, and actually describes a large num- ber of physically real systems. For example, a ve ...

where the potential energy is high, even if the potential energy is greater than the total energy of the particle. Furthermore, ...

of electrons to pass a very, very small gap between a sharp tip and a surface. Since the amount of tunneling varies exponentiall ...

Now we distribute the product XYZto all three derivatives in the Hamiltonian operator:  2 m ^2 (^)    x 2 2 XYZ (^)   y ...

In comparing these three equations with the original Schrödinger equation for this system, it is not too difficult to see that E ...