Title: ETH_Sem4 / PC3 / Woche03

Q: Born Interpretation der Wellenfunktion
A: \(\int\int\int_V |\Psi(x,y,z)|d\tau=P_v\)  \(P_V\) is the probability to find the particle in volume \(V\)

Q: What makes a wave function acceptable
A: 1 \(\int_{-\infty}^\infty \Psi(x)^*\Psi(x) dx< \infty\)
2 \(\Psi\) has to be single valued and a solution to the Schrödinger equation

Q: General formulation of the uncertainty principle for operators \(\hat{A}\) and \(\hat{B}\)
A: \(\Delta A=\sqrt{\langle\hat{A^2}\rangle-\langle\hat{A}\rangle^2}\)
\(\Delta B=\sqrt{\langle\hat{B^2}\rangle-\langle\hat{B}\rangle^2}\)
\(\Delta A \Delta B \geq \frac{1}{2}|\langle [ \hat{A},\hat{B}]\rangle|\)

Q: What is the variation principle and how do you use it (more information  Physikalische Chemie III: Molekulare Quantenmechanik U.Hollenstein und F. Merkt, 3.13, Das Variationsprinzip)
A: Take a fitting Test function (maybe a linear combination of orbitals if your doing chemistry or plain waves if your a solid state kind of guy or maybe polynomials if you are stupid)
Now since the lowest energy of the system is given by its actual eigenfunction for the given hamiltonian any other function will give you a larger energy value \[E_{test}\geq E_{eigen,1}\]Now just optimize the parameters in your test function to minimize this expresion (by doing the derivative of it and setting it to 0 and on and on and on, you passed math I&II, you got this)
\[\frac{\bra{\Psi}\hat{H}\ket{\Psi}}{\bra{\Psi}\Psi \rangle}\]

Q: Postulat 4: what is it, how can we use it?
A: The time evolution of a system with time independent Hamiltonion is modeled by the time dependent Schrödinger equation:
\[\hat{H}\Psi(q_i,t)=i\hbar \frac{d\Psi(q_i,t)}{dt}\]We solve this stuff with the Separtion of variables \(\Psi_n(q_i,t)=\psi_n(q_i)T(t)\)
and we find out that \(\psi(q_i)\) is the solution to the time independent equation and \(T(t) = e^{\frac{-iE_nt}{\hbar}}\)
this is stationary for a eigenstate (because the exponents cancel if we conjugate-square) it but if we take a linear combination of a bunch of them we can make it move (they dont cancel anymore as the \(E_n \) are all different we get the cross terms being time dependent).