Energy
Read Engel and Reid
Section 13.4
The Schrödinger equation
This section of the text derives Schrödinger's equation by assuming that atomic and molecular particles obey the time-independent wave equation, and that the wavelength of the particles obeys the de Broglie equation, λ = h/p.
Recall that we can write the total energy, E, as a sum of kinetic and potential terms:
Using p2 = (mv)2 = 2mK = 2m(E - V), we obtain p = [2m(E - V)]½, so that using the de Broglie relation λ = hp yields
| λ2 = | h2 |
| 2m(E - V) |
Substituting this into the time-independent wave equation (13.17) we get
| d2ψ(x) | + | 4π22m(E - V) | ψ(x) = 0 |
| dx2 | h2 |
which is readily rearranged to obtain the Schrödinger equation for a particle of mass m moving in one dimension with energy E:
| -h2 | d2ψ(x) | + Vψ(x) = Eψ(x) | |
| 8π2m | dx2 |
The Schrödinger equation can be rewritten in a more compact form:
| H | 2 | d2ψ | + Vψ = Eψ | |
| 2m | dx2 |
where H = h/2π (pronounced “h-bar”) and V and ψ are both functions of x. The first term in this expression can be identified with the kinetic energy of the system, and it has a similar form for all systems. But the potential energy term depends greatly on the system; for a free particle V = 0, and for a harmonic oscillator V = ½kx2
An interesting insight into the relationship between the form of the wavefunction ψ and the kinetic energy K can be obtained from
| λ = | h |
| (2m(E - V))½ |
| = | h |
| (2mK))½ |
Hence the greater the kinetic energy K, the smaller the wavelength. But we can relate the second derivative d2ψ/dx2 to the curvature of the wavefunction (i.e. the rate of change of slope). When a wavefunction is sharply curved (has lots of squiggles) the kinetic energy is large. When ψ is not so sharply curved (long wavelength, few squiggles) K is small. The association of curvature with kinetic energy is often a valuable clue to the interpretation of wavefunctions.
