Max Diem - Quantum Mechanical Foundations of Molecular Spectroscopy

Example 2.3

1 What is the probability P of finding a PiB in the center third of the box for n = 1?

2 What is P for the same range for a classical particle?

Answer:

1 The probability P of finding a quantum mechanical particle–wave is given by the square of the amplitude of the wavefunction. Thus,(E2.3.1)The integral over the sin2 function can be evaluated using(E2.3.2)Then the probability is(E.2.3.3)

2 A classical particle would be found with equal probability anywhere in the box; thus, the probability of finding it in the center third would just 1/3. Note that for higher values of n, the probability of finding it in the center third will decrease.

The principles derived in the previous section can easily by expanded to a two‐dimensional (2D) case. Here, an electron would be confined in a box with dimensions L xin the x ‐direction and L yin the y ‐direction, with zero potential energy inside the box and infinitely high potential energy outside the box:

(2.42)

The Hamiltonian for this system is

(2.43)

and the total wavefunction ψ x, ycan be written as

(2.44)

where A as before is an amplitude (normalization) constant. The total energy of the system is

(2.45)

Figure 2.4 Wavefunctions of the two‐dimensional particle in a box for (a) n x= 1 and n y= 2 and (b) n x= 2 and n y= 1.

For a square box with L x= L y= L , the energy expression simplifies to

(2.46)

The wavefunctions can now be represented as shown in Figure 2.4for the cases n x= 2 and n y= 1 and n x= 1 and n y= 2. These wavefunctions represent the standing wave on a square drum. Notice that the energy eigenvalues for these two cases are the same:

(2.47)

When two or more energy eigenvalues for different combination of quantum numbers are the same, these energy states are said to be degenerate . Here, for n x= 2 and n y= 1 and n x= 1 and n y= 2, the same energy eigenvalues are obtained; consequently, E 21and E 12are degenerate. This is a common occurrence in quantum mechanics, as will be seen later in the discussion of the hydrogen atom ( Chapter 7), where all the three 2p orbitals, the five 3d orbitals, and the seven 4f orbitals are found to be degenerate.

Next, the case of a system without the restriction of the boundary conditions (an unbound particle) will be discussed. This discussion starts with the same Hamiltonian used before:

(2.23)

When this differential equation is solved without the previously used boundary conditions

(2.29)

the new solutions represent a particle–wave that travels along the positive or negative x ‐direction. The most general solution of the differential Eq. (2.23)is

(2.48)

where b is a constant.

The second derivative of Eq. (2.48)is given by

(2.49)

with

(2.50)

or

(2.51)

Equation (2.51)was obtained by substituting

(2.52)

into Eq. (2.50). Thus, the unbound particle can be described by a traveling wave (as opposed to a standing wave)

(2.53)

carrying a momentum

(2.54)

into the positive or negative x ‐direction. kis the wave vector defined in Eq. (1.6).

Finally, the particle in a box with a finite energy barrier, V 0, will be discussed qualitatively. This is a situation where the particle is no longer strictly forbidden outside the confinement box and leads to the concept of tunneling, that is, the probability of the electron found outside the box. The shape of the potential function is shown in Figure 2.5b.

The potential energy for this case is written as

(2.55)

and

(2.56)