Is it possible to measure energy of \(0.75 \hbar \omega\) for a quantum harmonic oscillator? Why? Why not? Explain.

At the heart of understanding phenomena at the subatomic level is quantum mechanics, a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It challenges our classical views by revealing that energy, matter, and even time behave in counterintuitive ways.

One of the key features of quantum mechanics is the quantization of energy, which is particularly evident in systems like the quantum harmonic oscillator. Here, energy levels are discrete, meaning that the system can only occupy specific energy states. Unlike classical systems, where a swing can move smoothly to any height with the right push, quantum systems 'jump' between these levels with no in-between states. This is analogous to climbing a staircase: you can stand on one step or another but not in between the steps. This quantization leads to the fascinating conclusion that certain measurements, such as energy, can only result in certain discrete values, which has profound implications for our understanding of the microscopic world.

Energy quantization is a fundamental principle of quantum mechanics which states that certain physical quantities, such as the energy of a quantum system, can only take on specific, discrete values. Think of it as if energy levels were rungs on a ladder; an electron in an atom can only 'sit' on these rungs, not in the spaces between them.

This concept is clearly observed in exercises involving the quantum harmonic oscillator. The oscillator's energy levels are given by \(E_n = (n + \frac{1}{2})\hbar\omega\), where \(n\) can be any non-negative integer, depicting the 'rungs' on the energy ladder. Just as a person cannot stand at a height between two ladder rungs, a quantum harmonic oscillator cannot have an energy value that isn't defined by that specific formula. Therefore, encountering a value that isn't a non-negative integer multiple of \(\hbar\omega\), such as 0.75 in our example, signifies an impossible or non-existent 'rung' on this quantized energy ladder.

Planck's constant (usually denoted as \(h\)) is one of the foundational quantities in quantum mechanics, signifying the scale at which quantum effects become significant. Max Planck, who introduced the concept, uncovered this constant through his work on black-body radiation. The reduced Planck's constant, \(\hbar\) (pronounced 'h-bar'), which is equal to \(h/(2\pi)\), is often used in quantum mechanics and appears prominently in the expression for the energy levels of a quantum harmonic oscillator.

Having a fixed value of approximately \(6.626 \times 10^{-34}\) Js, Planck's constant marks the size of 'quanta' of energy. It effectively sets the smallest possible 'packet' of energy that can exist, making it a fundamental unit in energy quantization. In the context of the quantum harmonic oscillator, \(\hbar\) helps determine the minimum energy change possible when moving between quantized energy states. It is this granularity of energy that makes it impossible for certain values, such as the one given in the exercise (0.75 \(\hbar\omega\)), to exist as they do not meet the quantized 'rung' criteria established by this constant.