The Quantum Botanist

June 3, 2026

Quantum mechanics is generally considered a very difficult topic. To do anything meaningful or worthwhile, you need to do a lot of complex math, and a lot of things seem to go against what we think should occur based off of our classical world observations. There are four postulates of quantum mechanics, and they each have a classical equivalent. These are the basis that quantum mechanics works off of, similar to how $F=ma$ is a fundamental rule in classical mechanics.$\newcommand{\k}[1]{|#1⟩}$

Postulate Number	Classical Mechanics	Quantum Mechanics
I	The state of a particle at any given time is specified by the two variables $x(t)$ and $p(t)$	The state of a particle is represented by a vector $\newcommand{\k}[1]{|#1⟩} \k{\psi(t)}$ in a Hilbert space
II	Every dynamical variable $\omega$ is a function of $x$ and $p$: $\omega=\omega(x,p)$	The independent variables $x$ and $p$ of classical mechanics are represented by Hermitian operators $X$ and $P$ with the following matrix elements in the eigenbasis of $X$ (see equation 1 below). The operators corresponding to dependent variables $\omega(x,p)$ are given Hermitian operators$\Omega(X,P)=\omega(x\rightarrow X,p\rightarrow P)$
III	If the particle is in a state given by $x$ and $p$, the measurement of the variable $\omega$ will yield a value $\omega(x,p)$. The state will remain unaffected	If the particle is in state $\newcommand{\k}[1]{|#1⟩}\k{\psi}$, measurement of the variable (corresponding to) $\Omega$ will yield one of the eigenvalues $\omega$ with probability $\newcommand{\d}[2]{⟨#1|#2⟩}P(\omega)\propto\left\|\d{\omega}{\psi}\right\|^2$. The state of the system will change from $\newcommand{\k}[1]{|#1⟩}\k{\psi}$ to $\newcommand{\k}[1]{|#1⟩}\k{\omega}$ as a result of the measurement
IV	The state variables change with respect to time according to Hamilton’s equations (see equation 2 below)	The state vector $\newcommand{\k}[1]{|#1⟩}\k{\psi(t)}$ obeys the Schrödinger equation$\newcommand{\k}[1]{|#1⟩}i\hbar\frac{d}{dt}\k{\psi(t)}=H\ket{\psi(t)}$ where $H(X,P)=\mathscr{H}(x\rightarrow X,p\rightarrow P)$ is the quantum Hamiltonian operator and $\mathscr{H}$ is the Hamiltonian for the corresponding classical problem

Equation 1:

Equation 2:

These postulates may seem very complicated and confusing, especially with all of the terminology, but it is a lot less complicated than it seems.

First Postulate

In classical mechanics, the state of a particle is given by its position and momentum. In quantum mechanics, this is adjusted to a vector that contains the wave function of the particle. As for what a Hilbert space is, in this context, we are talking about a physical Hilbert space, where all functions can be normalized to one or the Dirac delta function.

Dirac Delta Function

The Dirac delta function is just the function where all values are 0, except one, which is infinite, denoted by $\delta$. This function is used to pick out a single value in an integral. For example, when evaluating the integral $\int^{\infin}_{-\infin}x\delta(3)dx$, the Dirac delta function would let us pick out the value at 3, so we would get 3. It can also be thought of as a really tall Gaussian curve.

Going back to the first postulate, all this means is that we are defining the state of a particle as a function that we are denoting as a vector.

Second Postulate

In classical mechanics, we can determine any variable using just the position and momentum (as well as any outside information that is not intrinsic to the particle, like the presence of a force field). In quantum mechanics, these variables are replaced by things called operators. Operators can be thought of as matrices that we use to measure different variables. For example, the position operator is used to measure the position of the particle. These operators are what give rise to the fact that measuring the position makes the momentum now unknown, as they do not necessarily commute, so doing one will change the results of doing the other. We can define any operator by combining the position and momentum operators in specific ways, similar to how we would with the classical position and momentum.

Third Postulate

In classical mechanics, measuring a variable does not necessarily change the state of a system. I can look at a ball, and the act of me looking at it to measure its position does not change where it is, or any other property it has. In quantum mechanics, however, measurement may change the state. When measuring with any certain operator, $\Omega$, the state will collapse into one of the eigenvectors of $\Omega$, and we will always get one of the eigenvalues, $\omega$, of the operator. This means that, by observing the state, we are actively changing it. This is where a lot of confusion usually starts. This would be like if you looked at a ball, and instead of just seeing where it was, it actually changed to be in a proper location as you looked at it. Quantum particles are in a state that is described by a wave equation, as we know from the first postulate. But, as we know from some of the core quantum experiments, like the double slit experiment, quantum “particles” are not always particles, they are also waves. When we observe them, quantum particles may swap between acting as a wave and as a parrticle. For example, when measuring position, we can only really measure it as a particle, at least if we want an accurate and precise answer. What would be the position of a wave? Think of something like a sine wave, and try to think of where it is. There is no good answer to that question. As the wave starts to get more centralized, we can start saying with more confidence where it is. For example, if our wave is just one spike up, we could reasonably answer that question. That is a more “physical” explanation of why that happens. A different explanation is that a quantum particle is probablistic in nature. Before observation, the particle does not actually have any value like position, it has a set of possible values with different probabilities defined by its wave function. When we observe it, we are forcing it to chose one of these values, therefore changing its state to one that has a 100% probability of having the variable you observed.

Fourth Postulate

This postulate goes back to being simple. State vectors all obey the Schrödinger equation. This is just a new equation that we are using to describe a quantum particle, with the classical equivalent being Hamilton’s equations. The only confusing parts would be that there are imaginary numbers and some notation that could be confusing, but as long as you just treat it like an equation and try not to get lost in notation too much, it should all work out properly.

Further Reading

For further information, I would highly recommend reading Shankar’s Principles of Quantum Mechanics (link on my book recommendations page). It goes over all of this in far greater detail than I do here.