@JohnRennie had a good trip sir hi

@JackRod Hi :-)

hi

no phones no internet

You've done your trip to the Dhikala zone?

Did you see a tiger? :-)

yes 5 with baby cubs

Wow!

So it was really worth it then!

one was so close to our open jeep that i could hear his breathing voice

we even watched the real hunting

when tiger just hunted a big sambhar dear

I must admit I am envious. I have only ever seen a tiger at a zoo.

ok sir actually i have some question related to quantum stuff are u free?

@JohnRennie

I'm helping someone with Python coding at the moment, but ask the question and I'll look at it as soon as I'm free.

@JohnRennie ok then, actually sir I was reading about Hilbert space, I am looking for a simple way to understand why do we need infinite-dimensional Hilbert spaces in physics, and when exactly do they become necessary: in classical, quantum, or relativistic quantum physics (i.e. when particles can be created and destroyed)?

Suppose we have a free particle, and for simplicity consider just one dimension.

ok

Then we solve the Schrodinger equation to find the eigenstates, and then we use these eigenstates as the basis for writing any arbitrary wavefunction.

i.e. if the eigenstates are $\psi_i$ then we can write any wavefunction as $\Psi = \sum a_i \psi_i$.

yes

This works because if we have a set of solutions to a linear differential equation then any sum of the solutions is also a solution to the differential equation.

So the eigenfunctions $\psi_i$ form the lements of a vector space.

That is, we take them as the unit vectors, then any vector $\Psi$ can be written as a sum of the basis vectors.

OK so far?

yes bsolutely

The dimensionality of the vector space is equal to the number of the eigenfunctions $\psi_i$. Yes?

i.e. the dimension is equal to the number of basis vectors.

sorry connection was lost

@JackRod Hi :-)

hi

The point is that there are an infinite number of the eigenfunctions $\psi_i$ because each eigenfunction corresponds to a specific momentum, and the momentum is a continuous variable.

That means the vector space has an infinite dimension.

The vector space is of course the Hilbert space for this system.