Quantum Mechanics

Quantum mechanics is the fundamental theory that explains the behaviour of matter and light and its unusual characteristics occuring at and below the scale of atoms. The way these tiny things act is very different from what we see in our everyday world. Because we can't easily see or directly experience these quantum behaviors, we use mathematical abstraction to understand them. Quantum mechanics provides a mathematical framework, of this quantum world. We start by learning the mathematical rules and tools of quantum mechanics. By working with this mathematical abstraction, we can predict how quantum systems will behave. While we might not intuitively "see" what's happening at the quantum level like we do with everyday objects, the math allows us to describe it accurately and build new theories.

The key connection between the actual quantum world and this mathematical description is made through the postulates of quantum mechanics. These are the basic rules that link the physical reality of quantum systems to the mathematical objects and operations we use to describe them.

Postulate 1: The State Space of a Quantum System

Statement: Associated to any isolated physical system is a complex vector space with inner product (that is, a Hilbert space) known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the system’s state space.

What does this mean?
Every quantum system has something called a state space. You can think of this as the space that includes all the possible states the quantum system could be in. According to the postulate, this space isn’t just any space—it has a very specific structure: it's a complex vector space with an inner product (this is what’s called a Hilbert space—for now, don't worry too much about the term itself).

What matters here is that:

It’s a vector space, which means each state is like an arrow or vector in this space.
It’s defined over complex numbers, not just real numbers.
Because it has an inner product, we can talk about the length of these arrows.

At any given moment, the complete physical condition of the object is described by a single point or arrow within this state space. This arrow is called the state vector. This state vector must be a unit vector—meaning its length must always be 1. (This is important, and we’ll explain why this condition is needed later.)

Technically, in quantum mechanics, what really matters is not the exact vector, but the direction it points in. So physically, the state corresponds to a ray, which is just the direction of the vector in space (not caring about multiplying it by some non-zero complex number).

In short , Every quantum system has a mathematical space of possibilities (its state space), and its exact state at any time is perfectly captured by a single, normalized arrow (its state vector) in that space.

Example: The Qubit

The simplest quantum system we study is called a qubit. A qubit has a two-dimensional state space.
Let’s say we have two special vectors called basis vectors for this space:

|0⟩
|1⟩

These two form an orthonormal basis—which means they’re perpendicular (orthogonal) and each has unit length (normal).
Any state a qubit can be in can be written as a combination of these two basic states:
|ψ⟩ = a|0⟩ + b|1⟩

Here, a and b are complex numbers. This just means that the qubit's state is like a blend or mix of |0⟩ and |1⟩, with complex weights.

Since the state vector must be a unit vector, we require:
⟨ψ|ψ⟩ = 1,
which leads to:
|a|² + |b|² = 1

This ensures the state vector has length 1, as required.

At this stage, just think of the qubit as an abstract object. Don’t worry yet about how it’s built or physically realized.

Postulate 2: Unitary Evolution of a Closed Quantum System

Once we have seen where a quantum state lives and how it looks we need will study how it evolves with time.

Statement:
The evolution of a closed quantum system is described by a unitary transformation. That is, the state |ψ⟩ of the system at time t₁ is related to the state |ψ′⟩ at a later time t₂ by a unitary operator U (which depends only on t₁ and t₂):

|ψ′⟩ = U|ψ⟩

This postulates gives a description of how the physical state of a quantum system changes over time. It talks about the evolution in a closed quantum system, but what is a closed system.
A closed system is one that is completely isolated and does not interact with anything else—not with its surroundings, not with any other system - No energy is being added or removed in a way that isn't considered for within the system's own description, and no information is being lost to the surroundings.

And the postulate assures that in closed system, the evolution from one time to another can be described by applying a unitary operator. Just like quantum mechanics doesn’t tell us what the state of a system is or what its state space looks like (those are details we have to find out separately), it also doesn’t tell us which unitary operator applies to a particular system. All it tells us is: if the system is closed, its time evolution will be unitary.

A unitary transformation (performed by a unitary operator U) is a very special kind of transformation in the Hilbert space. Mathematically, a unitary operator U is one whose inverse (U−1) is equal to its adjoint (U†). This property, U†U=UU†=I (where I is the identity operator), has profound physical implications:

Preservation of Probability: Unitary transformations preserve the norm (or length) of the state vector. Remember from the first postulate that the state vector is a unit vector (length 1), and the square of the components of the state vector relates to probabilities. By preserving the norm, a unitary transformation ensures that the total probability of finding the system in any possible state remains 1 over time. The evolution doesn't create or destroy probability; it just redistributes it among the possible states.
Preservation of Inner Products: Unitary transformations preserve the inner product between any two state vectors. This means that the relationship between different states is maintained as the system evolves (if they are orthogonal they remain orthogonal). This is important for how probabilities of transitions between states behave.
Reversibility: Because a unitary operator has a well-defined inverse (U†), the evolution described by a unitary transformation is always reversible. If you know the state at t2, you can apply U† to get the state back at t1. This deterministic and reversible nature is a key feature of the evolution of closed quantum systems.

We know that in real life, almost nothing is truly closed (except, maybe, the entire universe). Most systems interact, even just a little, with their surroundings. But there are many systems that we can treat as approximately closed, where this postulate still gives a good description.

Even open systems can sometimes be seen as part of a larger closed system. For example, if we include the system and its environment together, that bigger system may evolve unitarily. A gain there is an issue here we said earlier that evolution can be described as applying unitary operator, But doesn't that mean we are doing something to the system from the outside? Wouldn't that make the system not closed. There is a solution to this which we will discuss after giving a refined version of 2nd postulate.

There is also a refined version of this 2nd postulate which describes evolution of quantum state in continuous time.

Postulate 2′: The time evolution of the state of a closed quantum system is described by the Schrodinger equation:

Here:

ℏ is Planck’s constant.
H is a special operator called the Hamiltonian,
|ψ⟩ is the quantum state.
The equation tells you how |ψ⟩ changes with time.

This is known as the Schrödinger equation. It is the foundation for how we describe time evolution in quantum systems. According to this differential equation if we know tha Hamiltonian of the system ( to get a glimpse of what a hamiltonian - A hamiltonian in QM like in classical mechanics is the mathematical object representing the energy of the system.) For a physical problem it is very difficult to figure out its hamilonian but that is no our goal here, so dont worry how we get it.

The Hamiltonian is a Hermitian operator, which means:

All its eigenvalues are real numbers (no imaginary parts).
Its eigenvectors form an orthonormal basis (meaning they are independent and normalized).

In physical terms:

The eigenvalues E of the Hamiltonian represent possible energy levels of the quantum system.
The eigenvectors |E⟩ corresponding to those energies are called energy eigenstates or stationary states.

And since the Hamiltonian H is a hermitian operator it has a Spectral Decomposition.

Where E's are eigenvalues and |E⟩'s are their corresponding normalized eigenvector (remember they form an orthonormal basis of the space). The lowest energy is known as the ground state energy for the system, and the corresponding energy eigenstate (or eigenspace) is known as the ground state. The reason the states |E⟩ are sometimes known as stationary states is because their only change in time is to acquire an overall numerical factor,

Consider an example: Let’s say the Hamiltonian is:

Now we try to understand the connection between the the unitary operator of postulate 2 and the hamiltonian in its refined version. First we solve the differential equation in the refined version of 2nd postulate and get:

One can easily verify that this U is an Unitary operator.

Before moving forward I would like to give a detailed proof of the,
Spectral Decomposition Theorem: Any normal operator M on a vector space V is diagonal with respect to some orthonormal basis for V . Conversely, any diagonalizable operator is normal.

The theorem says that if an operator M is normal (ie. it commutes with its adjoint, (MM†=M†M)), then there exist an orthonormal basis for V such that M is a diagonal matrix with respect to this basis. And conversely if an operator M can be diagonalized with respect to some orthonormal basis, then M must be a normal operator.

Spectral Decomposition Theorem

The converse part of Spectral Decomposition Theorem is easy to prove, just write the operator as:

(We can do this because operator M is diagonalizable with respect to an orthonormal basis {∣i⟩} .) Just write the operator as mentioned above and evaluate MM† and M†M.

I have given a detailed proof of the spectral decomposition theorem because later we will be using this to derive some results and it is a very important theorem simply from maths perspective.

Conclusion

In the real world, almost nothing is truly closed (most quantum systems aren’t completely isolated). They interact—at least a little—with their surroundings. But sometimes, we can ignore those tiny interactions and treat the system as if it were closed, which lets us use the usual rules of quantum mechanics, like unitary evolution.

Even if a system isn’t closed, we can often include its environment and the combination of the system and its environment may be treated as a larger, closed system, which does evolve in a unitary way (i.e., follows the Schrödinger equation).

In quantum computing, we often talk about applying a unitary gate, like the Pauli-X gate, to a qubit. But wait—that suggests there’s someone (us!) interacting with the qubit. Doesn’t that mean the qubit is not a closed system?

This seems like a contradiction, because earlier we said unitary evolution only applies to closed systems.
Example: A laser is focused on an atom. If we consider the full system—atom plus laser, it can be described by a detailed Hamiltonian. When we zoom in and look only at the atom, its behavior is almost, but not exactly, like a closed system.

Here’s the interesting part: the atom appears to evolve as if it had its own Hamiltonian. And this Hamiltonian contains variables like the laser’s intensity—things we can control from the outside. So, even though the atom isn’t closed, its evolution looks like it's being governed by a Hamiltonian that we can tweak. More generally, in experiments, we often use time-varying Hamiltonians—that is, Hamiltonians that change over time because we’re adjusting knobs or settings (like laser strength, magnetic fields, etc.). Even though these systems are not perfectly isolated, their evolution often still follows the Schrödinger equation, just with a Hamiltonian that depends on time. And that’s usually good enough for predicting how they behave. So, even though many quantum systems aren’t perfectly closed, we often approximate them as closed and describe their evolution with unitary operators.

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