Making sense of the first axiom of quantum mechanics- Part 1

First Axiom of Quantum Mechanics

With every quantum system, We associate a hilbert space $( \mathcal{H},\hspace{0.05cm} + ,\bullet , <|>)$
Pure states of the system can be associated with elements of $\mathcal{H}$ .

It's a set whose elements represent the states of the system.

The state of a system is the information needed in order to predict it's future.
In classical mechanics, the state is given by:
the position

x(t)

and velocity

\dfrac{dx}{dt}

How do we know that states in classical mechanics are given by x(t) and

\dfrac{dx}{dt}(t)

F= m\dfrac{d^{2}x}{dt^{2}}

Equations of motion are generally a second order differential equation, which requires two initial conditions: initial position x(0) and the first derivative

\dfrac{dx}{dt}(0)

That's the notion of classical state. However, for a quantum system (note: A system for which quantum effects become significant.), we cannot carry over the idea of a state described by $\left(x(t),\dfrac{dx}{dt}(t) \right)$ .

Why not?

Student

Why can't we define the quantum state by $\left( x(t) , \dfrac{dx}{dt} \right)$

Teacher

Usually, quantum effects shows up at very small scale. So, when you try to measure the position of a quantum system , you have to interact with it experimentally, for example, by shining photons at it, to observe it.
That interaction, will affect the trajectory and hence the velocity of the particle becomes unkonwn.

So, we cannot measure the position and the velocity at the same time. (Heisenberg uncertainty principle )

The impossibility of measuring poition and velocity at the same time.

When we analyse the double-slit experiment, we realise that the only thing we can hope to predict about the future is the probability of finding the quantum system within a volume v around position x. This is the notion of probability density. (note: because space is a continium, the probability to pin down a particle at an exact given position is always zero: one of infinitely many possible positions. So, what makes sense to ask, then, is the probability of finding the particle within a given volume, which is the idea of proability density.)

The height of the interference pattern is proportional to the probability density of finding the particle at small volume in the neigherhood of

x

.

Because quantum systems are not localised, they behave like waves in this sense: the quantum state is described by a whole function

\psi(x)

whose modulus squared gives the probability density ,

|\psi(x) |^{2}

We now know that quantum states are represented by functions

\psi

. But why should the state space have a Hilbert space structure?

Interference patterns add up, so we should be able to add wave functions:

\psi + \phi

We also need to scale the wave function by a real scalar

\lambda

in order to normalise the probability.
So, a vector space structure is what's needed to model the state space.

Since probabilities are related to

|\psi|

, defining an inner product from which a norm is a defined is the structure that we require.
Quantum state space is then given by the structure

(\mathcal{H}, + , \bullet , < | > )

We have seen that a Hilbert space provides a suitable mathematical structure for modelling the state space of a quantum system.

But what law governs the time evolution of a quantum state, analogous to $F=ma$ in classical mechanics? and what is the intuition behind time evolution in quantum mechanics?

Making sense of the first axiom of quantum mechanics- Part 1

First Axiom of Quantum Mechanics

Questions and ideas

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