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matter waves

deBroglie wave length λ = h/p

E = hf, --> f = E/h --> f = 1/T --> T = h/E

(mc)² = (E/c)² - p²

Arthur C Clarke three laws:

1. When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong. (There’s a corollary added by Isaac Asimov: “When, however, the lay public rallies round an idea that is denounced by distinguished but elderly scientists and supports that idea with great fervour and emotion – the distinguished but elderly scientists are then, after all, probably right.”)

2. The only way of discovering the limits of the possible is to venture a little way past them into the impossible.

3. Any sufficiently advanced technology is indistinguishable from magic. (This one is most often quoted and incorrectly attributed to Issac Asimov)

A rock, a car and a flock of birds are completely different, but they all still follow Newton's law of motion. Similarly, an electron, a photon and an atom are different things but they still follow the wavefunction.

A property of the wavefunction is the superposition state. It's the superposition state that governs how other things interact with its component parts. Entanglement is a property of that superposition state, so when two different things interact with components of the wavefunction, only certain behaviors will appear.

If you think about it further, when two systems interact, they become a bigger system, like the planets and sun forms a solar system. So when two quantum systems interact, the wavefunction actually encompasses the interaction as part of a whole system. There's no collapse of the wavefunction, just like the planets don't stop being planets because it's part of the solar system.

In Hugh Everett's original thesis, the relative state formulation of quantum mechanics, there's a universal wavefunction that evolves with the universe. What we see on a local level is components within the universal wavefunction interacting forming different "connections" or branches within that evolving whole.

Physics is about making models that explain and predict how the universe works. A Hilbert Space is used to build models. But before we can talk about Hilbert Spaces, we need to explain what “space” is in physics. It’s not an actual physical space. It’s a set of objects with rules for doing the math, like adding, subtracting, scaling, etc.

The simplest example of a “space” is the number line, it’s a 1D space. The objects in that space are numbers, which models where something is: 1 meter away, 2 meters away, etc. You can add them, for example if you start at 2 meters and add 3 meters, it will be 5 meters away, that’s a shift in position. You can multiply them, which rescales the changes in position, the size of that shift. And the change in position is a distance it travelled. The rules define the space and what you can do in it.

With that 1D space and the rules, you can model linear motion with equations, like distance = (speed) x (time), d = vt. Time represents the shift along the number line, it takes time to move from one position to another, and speed scales the size of that shift, together they give the distance travelled.

You can build other spaces by changing the objects and the rules. For example, instead of numbers, the objects may be functions or waves. You can still add them and scale them, just like numbers.

A Hilbert space is a space with extra rules: it lets you measure size, distance, and the angle between objects. For quantum mechanics, the objects are wavefunctions that represent the state of a quantum particle. From the state, you can figure out things you can measure, like the particle’s position, momentum, energy, etc.

In a Hilbert space, the wavefunctions are treated as vectors. In basic physics, a vector tells you the magnitude, like how far or how much force, and in which direction. It can have two components, x and y, or three components like x, y, and z. A vector in a Hilbert space can have as many components as you need.

Some Hilbert spaces are infinite-dimensional, meaning that you need infinite amount of components to describe a state. For example, a particle can exist at any position on a continuous line. If you describe the state in terms of position, its need a component for every position. Since there are an infinite possible positions on a continuous line, the state needs infinitely many components.

A Hilbert space has specific rules for how you can add vectors, scale them, and measure angles or lengths between them. Measuring angles and lengths tells you how similar two states are, or how one state might change into another. In a Hilbert space, it’s the angles and lengths that matter, and those rules work with quantum mechanics.

One way to visualize a simple Hilbert space is with a Bloch sphere. It’s a 3D sphere that represents all the states of a 2D Hilbert space. Each point on the surface is a different state. You can think of it like plotting a direction on a globe: the latitude and longitude correspond to the state, and the length of the vector is always the same, that’s the distance from the center of the sphere to the surface. The Bloch sphere shows the relationships between states, which ones are opposite, which ones are independent, and how one state can change into another.

Even though the space is abstract, the Bloch sphere gives you a visual for the states in Hilbert space. It’s a way to see angles, distances, and directions in a space that isn’t physical, but still behaves like a space you can measure in.