Quantization of Energy

Quantum mechanics provides the foundation principles for all molecular processes while molecular statistical mechanics builds upon that foundation, and itself serves as the basis for thermodynamics. Here we’ll develop thermodynamics in a molecular context. Molecules behave quantum mechanically which means that we will need to know some results that are derived from quantum mechanics.

Energy is quantized 

So what it means when we say that energy is quantized? To explain this let’s take an example of hot tea getting cold. Imagine if you will that your tea could have only certain temperatures.

Figure 1 - Temperature vs. time and energy levels

In previous figure on the left we have temperature vs. time diagram which shows how hot tea over time is getting cold and eventually drops at ambient temperature. If we look from macroscopic point of view (this is red curve on temperature vs. time diagram) we see that transition from hot to cold is continuous the curve is smooth. But this is not the case because transition from hot to ambient temperature is more staircase shaped function. Only certain temperature levels are allowed. So if we make a connection between temperature and energy that would imply that there are certain levels of energy that are allowed and levels in between are not. 

Relative Energy Spacing Versus Size

Fiugre 2 - Macroscopic point of view 

As you can see from Figure 2 that between starting and ending temperature there are huge number of temperature/energy levels we can approximate these energy levels by smooth curve as seen in previous figure. At microscopic level or level of atoms and molecules they have large relative energy spacing, so we must consider quantized energy levels. 

Figure 3 - Microscopic point of view

Energy is Quantized by h

Max Planck suggested that radiated energy can come only in quantized packets of size hv. So this is explanation of energy quantization by h or Plank’s constant. The energy can now be written in the following form: 

$$E=h\nu $$

where E is energy in Joules, v is frequency in Hz or s^-1 and h is Planck’s constant which is equal to:

$$h=6.626\cdot {{10}^{-34}}\text{Js}$$

We can thus specify the energy by any one of the following:

The frequency v(for example Hz or s^-1):

$$E=h\nu $$

The wavelength λ ( for example, m or nm): 

$$E=h\frac{c}{\lambda }$$

This is with use of the formula:

$$v=\frac{c}{\lambda }$$

The wavenumber (for example in cm^-1 or m^-1):

$$E=hc\tilde{\nu }$$

1-D Schrödinger Equation 

Schrödinger used wave mechanics to describe the properties of energy and matter. The 1-D Schrodinger equation can be expressed as: 

$$\underbrace{\frac{{{\hbar }^{2}}}{2m}\frac{{{\partial }^{2}}}{\partial {{x}^{2}}}{{\psi }_{n}}\left( x \right)}_{\text{Kinetic Energy}}+\underbrace{V\left( x \right){{\psi }_{n}}\left( x \right)}_{\text{Potential Energy}}=\underbrace{{{\varepsilon }_{n}}{{\psi }_{n}}\left( x \right)}_{\text{Allowed Total Energy}}$$

where: 

$$\hbar =\frac{h}{2\pi }$$ 

$${{\psi }_{n}}\left( x \right)-\text{Wave function}$$

Solving the Schrodinger equation for a given potential V and set of boundary condtions yields a set of wave functions ψ_n(x) and a set of associated energies ε_n that are said to be ALLOWED. The integer index n specifies the state. Defining and solving relevant Schrodinger equation is the subject of quantum mechanics. Here, however, we need to be familiar only with allowed energy levels for the system that we will encounter.