A quantum mechanical oscillator vibrates at a frequency of \(250.0 \mathrm{THz}\). What is the minimum energy of radiation it can emit?

Planck's formula is a fundamental principle in quantum mechanics that relates the energy of a photon to its frequency. Named after Max Planck, who proposed the idea, the formula is given by:
\[ E = h u \]
where \( E \) is the energy of the photon, \( h \) is Planck's constant (6.626\times10^{-34} \, \mathrm{J \cdot s}), and \( u \) (Greek letter nu) is the frequency of the electromagnetic radiation.
This equation was revolutionary because it introduced the concept of quantization of energy, suggesting that energy is not continuous but comes in discrete 'packets' called quanta. In the context of the exercise, Planck's formula allows us to calculate the minimum energy that a quantum mechanical oscillator can emit when it vibrates at a certain frequency. This discovery led to the development of quantum theory, which has profound implications for understanding the nature of atoms and light.

When dealing with electromagnetic radiation, it's quite common to encounter different units of frequency. Converting frequency from one unit to another, such as from terahertz (THz) to hertz (Hz), is essential for making appropriate calculations. A frequency of 1 THz is equivalent to \( 10^{12} \) Hz. So, to convert terahertz to hertz, you multiply by \( 10^{12} \), and to convert in the other direction, from hertz to terahertz, you divide by \( 10^{12} \).
In the given problem, we had to convert the oscillator's frequency of 250.0 THz into hertz before we could apply Planck's formula. The frequency conversion step is critical because Planck's constant is given in terms of joule-seconds, and we need the frequency in hertz to ensure consistent units when calculating the energy. Remember, keeping track of units is crucial in physics to make sure your final answer makes sense both numerically and dimensionally.

The minimum energy of radiation that can be emitted by an oscillator is significant in quantum mechanics. This minimum energy corresponds to the smallest 'packet' or quantum of energy that the system can release. According to Planck's formula, this energy is directly proportional to the frequency of the radiation—the higher the frequency, the greater the minimum energy.
For an oscillator operating in the quantum realm, such as an atom's electrons transitioning between energy levels, this minimum energy is what is radiated away when the oscillator makes a transition down to its lowest energy state, also known as the ground state.
By applying Planck's formula as shown in the solution for the exercise, we can determine the quantum mechanical oscillator's emission. It is important for students to understand that this isn't arbitrary; it's a physical representation of a fundamentally discrete nature of energy at small scales, which classic physics couldn't explain before the advent of quantum mechanics.