 When you learn quantum mechanics you inevitably come across a thought experiment known as particle in a box. Although that's got nothing to do with shodding as cat in a box, it's still an integral thought experiment and model for learning about the nature of quantum mechanical systems and it can even be used as a starting point for predicting the energies of real atoms and molecules. In short, we have a box and to simplify the maths a little bit we make it one-dimensional. Just go with that. We drop a particle into this one-dimensional box and we allow it to move completely freely. It can move how it likes and where it likes but it can't leave the confines of that 1D box. In quantum mechanics the particle's probable location is determined by a mathematical expression known as a wave function. And we can figure out that wave function by looking at the constraints of the system. For instance the function must be zero at both of the extreme edges of the box. We apply a relatively straightforward mathematical operator to that function and out pops an expression for the particle's kinetic energy. But the important result that we are interested in is that that energy is quantized. The particle's kinetic energy can only take a certain discrete energy values with nothing in between. And each of these so-called energy levels is related to a single number known as a quantum number. When we talk about energy being quantized in quantum mechanics it's often treated as if it's a strange and mysterious thing like a car going at one speed and then another speed without ever passing through any intermediate velocity. But I want to show you one interpretation of this where we actually expect quantization and it absolutely reflects the 1D particle in a box model. When you play a note on a musical instrument you expect the same note to appear again and again. Music would be incredibly difficult if there was any randomness to this. And on a stringed instrument like a guitar you can quite easily see why that's the case. At one end we have a nut and the other end a bridge and both of these fix the string in place and the string must vibrate in such a way that these points always remain fixed. So is that starting to sound similar to one of our particle models yet? If we look very closely at a vibrating string we see the peaks in the middle. The maximum displacement is here and it's stationary at the end. Now the sound produced is controlled by a number of factors. There's the weight and size of the string and the tension but mostly it's the length. For a given weight and size and tension of a string only certain frequencies satisfy the other constraints. So if we reduce the length of the string the pitch or sound herd changes. This is analogous to changing the length in our wave function equation. The exact sound you hear however is a little more complicated. There are various harmonic frequencies vibrating through the string at once although you can dampen them out if you pluck very carefully in the middle and get this almost synthetic sounding tone out that's a pure ish sine wave. Pluck more closely to the bridge and you get a much more tinny sound as the higher frequencies get thrown in there. But the note remains the same. But that's not all. We can change the pitch of the string without doing anything else to it. We don't need to shorten it or tighten it. We simply need to touch lightly in the right location and there you have it. The first natural harmonic. The string length is the same it's tightness and mass is the same but it's now an octave higher. What's happening here particularly if we look very closely is the shape of the vibration has changed. It really is stationary in the middle and its maximum displacements are a quarter and three quarters of the way along the string. That means that the wavelength has halved the frequency has doubled the sound has gone up an octave and this is possible precisely because that shape of vibration still satisfies the constraints that it must be fixed and zero at the edges. But there is more. We can do this a third of the way along. Not quite doubling the frequency again but this one is curious because the string really does have two stationary points and maximum displacements at three. We can't replicate the trick just anywhere however because these waves don't fit they don't satisfy that constraint. Incidentally this is why perfect fifths sound well perfect. The peaks troughs and nodes of the waves line up pretty nicely. When the wave does this in the particle in a box model the quantum number has increased by one and the energy has increased only certain wavelengths fit into the box only certain frequencies fit into the box and because energy is related to frequency the particle can only take certain energies. So the idea of a particle in a box isn't that crazy or unexpected much of what we think about when we're told about quantum weirdness really falls out of using these concepts of waves of frequencies and vibrations and what constrains them but applying them in a slightly different context.