So I want to really learn QM, and master it. I wanted to do it in a bit of a different way, by asking questions and stating what i think is going on and havuning a conversation where I will be corrected. shall i start, from the very basics? The hilbert space
so tell me where I am wrong in the line of thinking
I want to think it then formalize it with your help, as step one
ok for the thinking
part
So it is a space, infinitely large. In some sense it is like a euclidean space but with $x_1 to x _{infinity}$, but the length must be less than infinity. Is this true?
All hilbert spaces have a hilbert basis, which is a countable linear combination of vectors such that the space generated by this basis is dense in the hilbert space but not necessary complete
Actually I made a small mistake. Some hilbert spaces don't have a countable hilbert basis (these are known as non separable hilbert spaces), and also orthonormal is necessary for something to be a hilbert basis
It is. Given a hilbert basis $\{\lvert e_i\rangle\}$, it is orthonormal, therefore for any ket $\lvert x \rangle$, the inner product $\langle x\vert e_i\rangle = x_i$ are the ith components of $\lvert x \rangle$
That I am not terribly sure. Measure theory is one area I am not good at and I already made many mistakes previously from irrationals to cantor sets and so on
You will be better off asking 0celo or acuriousmind about measure theory stuff
You can just stick to standard Riemann integrals, there's no need for Lebesgue integration in this problem.
also, you've overloaded your use of $x$
if we work in 1-dimensions, then $P=\int_a^b |\psi(x)|^2 dx$ is interpreted to be the probability of finding the particle represented by wave function $\psi$ within the interval $[a,b]$
The moment of inertia is a measure of the distribution of mass in a system. Specifically how it's distributed with respect to a central axis of rotation.
The larger the moment of inertia, the harder it is to get that system spinning. In the real world, that corresponds to more mass being distributed farther away from the central axis.
Spin is a very deep quantum mechanical concept and I'm not prepared to give a clear definition lol.
@goodnight alright, lets think about spin statistics theorem, although it would be much more fun if we knew what spin was to some rudimentary degree of sophistication
lol
@enumaris what is a spin quantum number (not spin lol)
Stern gerlach is indeed the first experiment that demonstrate the existence of spin as an extra degree of freedom in a quantum system. I don't recall other details except some of the experiment set up and results. But since our focus here seemed to be mathematical formulation, we can safely ignore the experiment details until later
Also I don't know the derivation of the spin statistic theorem, though I know it's consequences
I feel like a should try to solve the quantum harmonic, and see what happens before I take the next step. You will walk me through stating and solving this right? hehe
I want to do this so that i can use some raising and lowering operator trick later
I know it looks like we are navigating blindly but I promise there is purpose to this :D
Ok let me try to think of the harmonic oscillator with no damping
Because if you don't know what step that is, solving this equation $H\psi = E\psi$ doesn't get you anywhere other than saying you solved a math problem
And we retrieve the Time-independent Schroedinger's equation $H\psi=E\psi$, this gives us a static solution $\psi(x)$ which is time independent. In order to go back to the full solution, we append a $e^{iEt}$ factor like so: $\Psi(x,t) = \Sigma_n a_n \psi_n(x)e^{iE_n t}$
So, actually when we solve a problem, what are we doing? We are given an initial wave function $\Psi(x,t_0)$ and it evolves according to the (time dependent) Schroedinger's equation. How do we figure out this evolution? We solve the time-independent Schroedinger's equation for a set of energy eigenstates $\psi_n(x)$
According to the spectral theorem, the energy eigenstates will be $complete$ in that any (smooth) function $\Psi(x,t_0)$ can be expressed as a superposition of the eigenstates: $\Psi(x,t_0) = \Sigma_n a_n \psi_n$
Given any $\Psi(x,t_0)$ we first express it in terms of the eigenstates $\psi_n$ by figuring out the expansion coefficients $a_n$, and then the time evolution is trivial.
There's one last detail, which is "given $\Psi(x,t_0)$ and $\psi_n(x)$ how do we find the expansion coefficients $a_n$? The answer to this is by exploiting the $orthogonality$ and normalization of eigenfunctions, i.e. that $\int_{-\infty}^{\infty} \psi_n(x)\psi_m(x) dx = \delta_{nm}$ where $\delta_{nm}$ is the Kronecker delta that equals 1 when n=m and 0 when n != m.
