• Simultaneous determination of exact position and exact momentum of a particle is impossible
  • $\Delta x$ uncertainty in position
  • $\Delta p_x$ uncertainty in momentum
  • Uncertainty of an electron
  • $\Delta x \cdot \Delta p_x \geqslant \frac{h}{4 \pi} \Rightarrow \Delta x \cdot \Delta v \geqslant \frac{n}{4 \pi m}$
  • $\frac{6.626 \times 10^{-34} \mathrm{J} . \mathrm{s}}{4 \times 3.14 \times 9.1 \times 10^{-31} \mathrm{kg}}\approx 10^{-4} \mathrm{m}^2\mathrm{s}^{-1}$

• $\Delta x {.} \Delta v_x$ $\approx$ $10^{-4} {m^2}s^{-1}$

• $\Delta x {.} \Delta v_x \geqslant \frac{h}{4 \pi m}$
• When m = 100 g
• $\frac{6.626 \times 10^{-34} \mathrm{J} .5}{4 \times 3.14 \times 0.1 \mathrm{kg}} \approx 10^{-33} \mathrm{m}^2 \mathrm{s}^{-1}$

• $r_n$
• $v_n$
• Exact position and exact momentum

• Wavefunction $(\Psi,\psi)$
• Wavefunction can be
  • $x, x^2, \sin x, e^{-x}, e^{-i x}$, a+ib
  • For every quantum mechanical system their exist a wavefuntion
  • $\Psi$ - does not have physical meaning

• Wavefunction $(\Psi,\psi)$
  • According to max born $|\psi|^2$ has physical meaning
  • Probability of finding electron at a given point in space $|\psi|^2$ is always positive
  • $\psi=x \Rightarrow|\psi|^2=x^2$
  • $\psi =\left.e^{-x} \Rightarrow |\psi\right|^2=e^{-2 x}$
  • If $\psi$ is complex function then $|\psi|^2=\Psi^*\Psi$
  • $\psi=a+i b \Rightarrow|\psi|^2=\Psi^*\Psi$
  • $\psi=(a-i b)$ $(a+i b)=a^2+b^2$

• How to get information from $\psi$
• For every observable there exists an operator
• $+$ $\quad$ $-$ $\quad$ $\sqrt{}$ $\quad$ $()^2$ $\quad$ $\frac{d}{d x}$ $\quad$ $\int$ $\quad$ $\log$
• $\sqrt{9}=3$,$\quad$ $(9)^2=81$, $\quad$ ,$\frac{d}{d x}(a x)=a$
  • Position operator = $\hat{x}$
  • Momentum operator = $\hat{p}_x$
  • Energy operator = $\hat{H}$ (Hamiltonian)

• Schrödinger equation $\rightarrow$ $\hat{H} \Psi=E \Psi$
• $\hat{H}$ Hamiltonian
• $\Psi$ wavefunction
• E energy
• Unknown - $\Psi$ , E
• Known - $\hat{H}$

• Results from the solution of schrödinger equation
• $H \psi=E \psi$
• A series of wavefunction $\equiv$ orbitals
• $\underbrace{\text{atomic orbitals}}_ {\text{"address of electron"}}$
• A series of energies ${\text {(discrete set)}}$
  • {0,5,10,15}
  • {0,10,16,29}

• They tell us about size , shape , orientation
• Unique identification of orbital
  • Principal quantum number $(n)$
  • Azimuthal quantum number $(l)$
  • Magnetic quantum number $(m)$
  • Spin quantum number $\quad$ $\left(m_s\right)$

• Principal quantum number (n)

• It tell us about:

• Size of the orbital

• $n\quad$ = $\quad 1,\quad 2,\quad 3,\quad 4,\quad 5,..$

$\quad\quad\quad \downarrow$ $\quad\downarrow$ $\quad\downarrow \quad \downarrow$

• Shell = K$\quad$ L$\quad$ M $\quad$ N....

• Azimuthal Quantum Number
• Orbital quantum number (l) it tells us about the "shape" of the orbital
• $l$ = 0,$\quad$ 1,$\quad$ 2, $\dots$ , n-1

• $l$ = 0 $\quad$ sphere
• $l$ = 1 $\quad$dumbbell
• $l$ = 2 $\quad$double dumbbell

• Azimuthal Quantum Number
  • Orbital quantum number (l)
  • "Shape" of the orbital
  • $l$ = $\quad$ 0,$\quad$ 1,$\quad$ 2, $\dots$ , n-1

• $l$ = 0 $\quad$ sphere
• $l$ = 1 $\quad$dumbbell
• $l$ = 2 $\quad$double dumbbell
• $l$ = 3 $\quad$ $f$
• $n$ = $\quad$ shell
• $l$ = $\quad$ subshell