Quantum thoughts:

This writing is not going to look at the technical details of the topic, it is only the larger framework.

Everything tends to equilibrium. 

In quantum mechanics we always write the states. We work with the states and the operator. I think that’s pretty much of quantum mechanics. 

Mostly binary, 

Probability, states and operators. Now, all we need is to really understand what each term means.

Probability, 

This is a nice idea; I mean, to know the possibility of happening is actually amazing. After all, everything is probability. It has its value in between 0 and 1. 0 the least probable, and 1 the most. The probability is given as the ratio of favorable outcomes to the total outcomes. 

But the system might be too complicated and the distribution might be uneven, in which case we must learn to write the probability distribution function and then finding the probability.

So, this is a pretty remarkable thing. 

Anyway, all sorts of ways of calculating probabilities to be summarized in one line is: the ratio of favorable outcomes to the total outcome. 

States:

Or the eigenstates, an important quantum function’s state which has a very remarkable properties and benefits. 

State can generally be considered as a vector function in mathematics. All the quantum phenomenon are described by the quantum state of that particle. We must be able to write the Hamiltonian of the system which is the most basic idea although the hardest to figure out. 

So, we have a state, made up of the basis (Eigen)states. We can create a set of basis states in any operator’s corresponding eigenstates. However, the dimensionality of the states is same in any space. So, if we act any operator in its eigenstate then it will give it’s corresponding value known as the eigenvalue. 

This is how it works, hit the required operator in its state get its answer. 

Now, the most common eigenstates are position, momentum, and energy because these are the most needed physical values to calculate anything further. 

Interestingly, position and  momentum are something called as “canonical conjugate variables” so, they are the Fourier transformation of each other. Similar can be done for energy and time. 

Operators: These are the tools to get the value, for example ammeter is an operator to measure the current, like that, we will have a quantum mechanical operator corresponding to any of the physical measurement required. So, we have a position operator to measure the position, a momentum operator to measure the momentum, energy operator to measure the operator, and so many others depending on how much complex things you want me to say about quantum mechanics. So, operator acts on states giving the result.

Preliminary quantum procedure:

We have a quantum state. We can perform a linear translation, which is just the use of linear momentum operator. 

However, the state can have a rotational transformation. So, we need to introduce another operator known as rotational operator that rotates the state vector by some defined angles along some chosen axis. If the rotation along any axis at any angles leaves the state vector invariant then the state is called spherically symmetric. 

Now, rotation operator rotates the state just like in classical mechanics we have the rotation matrices, yeah just like them. This becomes angular momentum operator when it grows up. Then use this operator to get the value of angular momentum, and just a reminder, you can create the system in any of your operator’s corresponding basis. So, yes the angular momentum has its own basis. 

Then there are some others, like spin operator for the spin particles. This concept of spin is very similar to the one in classical physics, however, the reality and the math are different. Spin is an intrinsic property of the particles, it comes with the particle by birth. So, electrons, protons, neutrons, are spin 1/2 particles. There are other many particles with different spin numbers. Our interest mostly falls under spin 1/2 particles. So we apply spin operator to the state and get its value. 

All the operators are unitary to preserve the norm or the probability. We generally normalize the state vector to unity. 

The state vector satisfies Schrödinger’s equation, so that at any time we can find the corresponding state vector. 

An image of Schrödinger's equation