(in Q.M)
\(\ast\) Math is simpler than classical physics.
\(\ast\) Every thing at ultimate level turns out to obey Q.M.
\(\ast\) Q.M applies to everything i.e. photons, electrons, humans, galaxy. It is just that its manifestations become dramatic when we observe small object, & become more dramatic if small objects are moves with very high speed.
\(\ast\) The fact that two solid object does not penetrate one another is due to Q.M. principle (pauli exclusion principle).
\(\ast\) magnetism, superconductor, electric conduction, propogation of sounds in solids, these are all quantum phenomenas.
'The phenomena of diamagnetism, paramagnetism depends on quantum mechanical principales, as they are not explainable from classical physics.'
\(\ast\) Black body radiation is also explained from Q.M.
\(\ast\) uncertainty principle.
The position and canonical conjugate momentum of any particle can not be measured simultaneously and accuratly (to arbitrary precision).
uncertainty principle is intrinsic in nature :-
It is not lack of resolution / experimental lack, it is that the position and momentum are not defined to be measured to with arbitary accuracy (when measured simultaneously).
\(\ast\) we can see that hamiltonian formalism is translated to Quantum mechanics.
The whole assumption is that we have
-- Hamitonian and
-- Hamiltonian framework then we translate to Q.M.
Q.) It is very deep and harder question to ask how to quantize or how do we do Q.M. for a system which are not classical hamiltonian in classical limit.
e.g. friction in system.
(Q) "Problem of dissipation in Q.M is still an open problem."
classically, we know that
Q.M
in Q.M \(\delta x = \sqrt{\langle x^2\rangle - \langle x\rangle^2}\) is calculated w.r.t quantum mechanical probability distribution. and that distribution will be specified through the Shröndiger's equation.
For any two arbitrary canonically conjugate dynamical variable \(\exists\) uncertainty principle.
\(\langle [A,B]\rangle = \) expectation value of \([A,B]\).
we know that
\usetikzlibrary{arrows.meta,decorations.pathreplacing,calc,angles,quotes,positioning,patterns}
\begin{tikzpicture}[scale=1.0]
\draw[->] (-0.6,0) -- (5.2,0) node[below] {$\Delta x$};
\draw[->] (0,-0.4) -- (0,3.4) node[left] {$\Delta p_x$};
\draw[domain=0.55:4.6,smooth,variable=\x,thick] plot ({\x},{2.2/\x});
\foreach \a in {0.75,1.15,1.55,1.95,2.35} {
\draw ({\a},{2.2/\a}) -- ({\a+0.55},{2.2/\a+0.45});
}
\draw[dashed] (0,0) -- (0.9,2.45);
\node at (3.9,2.6) {$\left(\Delta x\,\Delta p_x \geq \tfrac{\hbar}{2}\right)$};
\draw[->] (4.3,0.15) to[out=-30,in=180] (4.9,-0.35);
\node[anchor=west] at (4.9,-0.5) {$\left(\Delta x\,\Delta p_x = \tfrac{\hbar}{2}\right)$};
\end{tikzpicture}
Now, the idea of phase space point and trajectory is no more in Q.M.
\usetikzlibrary{arrows.meta,decorations.pathreplacing,calc,angles,quotes,positioning,patterns}
\begin{tikzpicture}[scale=0.9]
\draw[->] (-0.9,0) -- (1.1,0) node[below] {$x$};
\draw[->] (0,-0.5) -- (0,1.3) node[left] {$p$};
\fill (0.35,1.0) circle (2.2pt);
\end{tikzpicture}
\usetikzlibrary{arrows.meta,decorations.pathreplacing,calc,angles,quotes,positioning,patterns}
\begin{tikzpicture}[scale=1.0]
\draw[thick] (0,0) ellipse (1.6 and 1.15);
\coordinate (P) at (0.95,0.93);
\draw (0,0) -- (P);
\node at (0.35,0.62) {$a_0$};
\draw[->] (P) -- ++(0.55,0.42) node[above right] {$p_\theta$};
\draw (0.5,0) arc (0:44:0.5);
\node at (0.72,0.22) {$\theta$};
\draw[->] (0.2,-1.14) -- (0.6,-1.06);
\end{tikzpicture}
Here, \(\theta, p_\theta\) can be measured with arbitary precision
so, we can not use orbits to describe motion of \(\bar e\) in an atom.
Also, Orbital theory is not completely correct.
\usetikzlibrary{arrows.meta,decorations.pathreplacing,calc,angles,quotes,positioning,patterns}
\begin{tikzpicture}[scale=1.0]
\draw[thick] (0,0) circle (0.85);
\fill (0,0) circle (1.2pt);
\draw (0,0) -- (0.85,0);
\node at (0.45,-0.22) {$a_0$};
\node at (2.6,0.55) {$n$};
\node at (3.5,0.55) {$\ell$};
\node at (4.4,0.55) {$m$};
\node at (2.6,-0.15) {$\underline{1}$};
\node at (3.5,-0.15) {$0$};
\node at (4.4,-0.15) {$0$};
\end{tikzpicture}
If any \(e^-\) is orbiting with radius \(a_0\), its angular momentum can not be zero. but it is told that
All objects in Q.M are treated with state vector which is generalization of wave function.
ex.
-- \(e^-\) is a state vector
-- photon is a state vector.
-- Even the molecular orbital theory is not correct \((s, p, d, f)\) [\(n\) orbitals also] are not correct.
"we should not extrapolate this categorisation of objects into waves & particles to the microscopic domain". It is conceivable that we have objects which have properties of both waves & particles, & it is also conceivable that \(\exists\) some objects which has properties of either of them, depending upon how we measure them."
The state of a quantum system is not specified by a point in phase-space but rather it is specified by an element of linear vector space. i.e. vector "state vector". \(\left(|\psi\rangle\right) \Big/ |\Psi(t)\rangle\).
\(\rightarrow\) This state vector is an element of LVS, and all the required information can be extracted from just state vector. and this state vector evolves in time.
\(\rightarrow\) we have to throw the idea of representing system as variation of many dynamical variables (independent), \(x_1, x_2, x_3 - x_n\) along with idea of phase space.
Contains set of elements ("vectors"). \(|\psi\rangle, |\phi\rangle, |\chi\rangle \in V\) with following properties / definition.
- \(|1\rangle + |2\rangle = |3\rangle \in V\) ; closure
- \(|1\rangle + \left(|2\rangle + |3\rangle\right) = \left(|1\rangle + |2\rangle\right) + |3\rangle\) ; Addition is associative
- \(|1\rangle + |2\rangle = |2\rangle + |1\rangle\) ; Add\(^{\text{n}}\) is commutative
- \(a|\psi\rangle \in V\) \(\hookrightarrow\) fields can be scalar Real/complex \((a, b, c, \cdots \in F(C \text{ or } R))\)
- \(a\left(b|1\rangle\right) = (ab)|1\rangle \in V\)
- \(a\left(|1\rangle + |2\rangle\right) = a|1\rangle + a|2\rangle\)
- \(\exists\) Null vector such that \(|1\rangle + |0\rangle = |1\rangle\)
- \(\exists\) an additive inverse such that \(|1\rangle + |-1\rangle = |0\rangle\) \[ \langle\phi|\psi\rangle^{*} = \langle\psi|\phi\rangle \] \[ \left(\langle\phi|\psi\rangle^{T}\right)^{\dagger} = \langle\psi|\phi\rangle^{*} \quad \Rightarrow \quad \langle\phi|\psi\rangle = \langle\psi|\phi\rangle^{*} \]
\(\hookrightarrow\) equals only when