entangledyouandme.com

Author: Surya Bhaskaram

  • Episode 3: Why is the Bohr Radius About 0.53 Å ?

    This is episode 3 of the series Why matter exists?

    If squeezing the electron is energetically expensive, shouldn’t the electron just keep moving farther and farther away?

    It would if kinetic energy is the only energy in the atom.

    Imagine an electron very far from proton. Now you try to bring it closer to proton. Does proton help you or oppose you?

    They are oppositely charged, so the proton helps by attracting the electron.

    So the system releases energy as they come together. That is why the potential energy becomes more and more negative.

    PE=−e24πε0rPE=-\frac{e^2}{4 \pi \varepsilon _0 r}

    Where does the released energy come from?

    From the potential energy of the electron–proton system. As the electric field does work on the electron, the system loses potential energy.

    I see… if the PE were positive instead… the electron would not remain bound to the proton and there would be no atom.

    But wait, bringing the electron closer also increases the uncertainty in its momentum.

    Exactly, greater momentum uncertainty means greater kinetic energy.

    KE=ℏ22mr2KE=\frac{\hbar^2}{2mr^2}

      (click here to see how KE=\frac{\hbar^2}{2mr^2}  )

    Even the graphs can teach more.

    Look at this graph:

    What do you notice?

    The kinetic energy shoots up very rapidly as the radius decreases.

    How does potential energy changes with radius?

    The PE decreases…

    Ummm….. But, as the electron comes closer, the potential energy should increase as it is proportional to 1/r …..right?

    Be careful, this is a very common mistake.

    The potential energy has actually decreased.

    The negative sign completely changes the interpretation.

    As the electron falls toward the proton, the electron–proton system releases energy and moves to a lower-energy state. That is why the potential energy becomes more negative.

    Never ignore the minus sign. It tells you the direction in which the energy is changing.

    I understand each graph individually. But the atom has both types of energy at the same time. So… which one does Nature actually care about?

    These graphs describe the two competing tendencies inside the atom. Nature doesn’t minimize kinetic or potential energy individually. It minimizes their sum, the total energy:

    E=KE+PEE=KE+PE

    This gives us the following graph.

    I can see the lowest point on the graph. But how can I find its exact value?

    Whenever a smooth curve reaches a maximum or minimum, its slope becomes zero,

    ∴dEdr=0\therefore \,\,\, \frac{dE}{dr}=0

    Ah… Solving the equation gives us:

    r=4πε0ℏ2me2r=\frac{4 \pi \varepsilon_0 \hbar^2}{me^2}

    And that’s the famous Bohr radius!

    Numerically: 0.529 Å.

    So the size of the hydrogen atom is not an arbitrary number.

    Exactly.

    It emerges naturally from the balance between quantum kinetic energy and electrostatic attraction.

  • Deriving Electron’s Kinetic Energy from Uncertainty Principle

    Say, electron is confined within a region of size ‘r’.

    The uncertainty principle tells us:

    ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}

    using Δx∼r\Delta x \sim r

    Δp≥ℏ2r\Delta p \geq \frac{\hbar}{2r}

    If the electron is confined to a region of size r, its momentum can not be smaller than its uncertainty in its momentum

    ∴p∼Δp\therefore p \sim \Delta p

    or,

    p≥ℏ2rp \geq \frac{\hbar}{2r}

    Using an order-of-magnitude approximation

    p∼ℏrp \sim \frac{\hbar}{r}

    substituting this estimate into classical expression for kinetic energy we get,

    KE=p22m∼ℏ22mr2KE=\frac{p^2}{2m} \sim \frac{\hbar^2}{2mr^2}

    KE=ℏ22mr2KE=\frac{\hbar^2}{2mr^2}

  • Episode 2: Can an electron ever sit inside the nucleus?

    This article is Episode 2 of the Satya–Chandra series Why Matter Exists.

    I’ve been thinking about our last conversation.

    You said the electron doesn’t collapse because the lowest allowed state is n=1

    But that doesn’t satisfy me.

    Why not?

    Because it feels like a mathematical trick.

    We simply wrote

    n= 1,\;2,\;3,\;\ldots

    And declared n\neq0

    Then we concluded the electron cannot reach the nucleus.

    Nature doesn’t work because of our equations.

    Our equations are supposed to describe nature.

    Why does nature forbid it?

    This is where modern quantum mechanics gives us a deeper answer.

    The electron is not merely a tiny particle. It also behaves like a quantum wave.

    According to Heisenberg’s uncertainty principle:

    \Delta x \, \Delta p \geq \frac{\hbar}{2}

    Here, \Delta x and \Delta p represent the uncertainties in the electron’s position and momentum, respectively.

    Now imagine if we confine an electron to a region as tiny as a nucleus, about   10^{-15}\,\mathrm{ m} across. That means, according to Heisenberg’s principle, such a tiny uncertainty in position demands an enormous uncertainty in its momentum and therefore an enormous kinetic energy.

    Satya:

    Enormous??? Can we actually calculate how impossible it is for the electron to stay inside the nucleus?

    Chandra:

    Certainly,

    \Large \Delta x \approx 10^{-15} \, \mathrm{m} \\ \\ \text{According to Heisenberg uncertainty principle:}\\ \\ \Delta p \geq \frac{1.055 \times 10^{-34}}{2 \times 10^{-15}}\\ \\ \Delta p \approx 5.28 \times 10^{-20} \mathrm{kg\,m/s} \\ \\ \text{Taking}\: p \approx \Delta p, \\ \\\ \text{the minimum kinetic energy is}\: K=\frac{p^2}{2m}=\frac{(5.28 \times 10^{-20})^2}{2 \times 9.1 \times 10^{-31}} \\ \\ K = 1.53 \times 10^{-9} \mathrm{J}\\ \\ K= 9.5 \mathrm{GeV}

    Whoo… that’s too high.

    Yes,

    to squeeze an electron into the nucleus, nature demands an enormous kinetic energy.

    Ah! So the electron doesn’t stay outside because the proton stops attracting it…it stays outside because squeezing it into the nucleus would require an enormous amount of energy!

    But one mystery still remains…

    If the electron cannot fall into the nucleus, why does it settle at one particular distance?

    The answer lies in one of the most remarkable numbers in physics—the Bohr radius.

    Episode 3: Why is the Bohr Radius About 0.53 Å ?

  • Series: Why matter exists?

    Episode 1: Why Doesn’t Matter Explode?

    Episode 2: Can an electron ever sit inside the nucleus?

    Episode 3: Why is the Bohr Radius About 0.53 Å ?

  • Episode 1: Why Doesn’t Matter Explode?

    This article is Episode 1 of the Satya–Chandra series Why Matter Exists.

    Electrons repel electrons. Protons repel protons. How is matter stable then?

    Oh! I see. It’s because a proton attracts an electron.

    But… if so, then why doesn’t electron fall into the nucleus?

    Umm… according to classical physics it should collapse.

    Yet, here we are.

    An orbiting electron is an accelerating charge. Accelerating charges radiate energy. Radiating energy means losing energy. Losing energy means spiralling inward.

    Ah…. Bohr…..

    Exactly, Bohr proposed something outrageous. He proposed that an electron could occupy only certain allowed orbits. Only those orbits whose angular momentum is an integral multiple of \hbar are allowed, i.e. L=n\hbar, n=1,\;2,\;3\;\ldots

    And remember, n\neq 0

    Therefore, there is no allowed orbit below the ground state.

    Oh! I see. That means an electron is not allowed to possess just any energy. Only certain discrete energy levels are permitted. It can jump only between these allowed levels.

    So there is a lowest allowed energy state n=1, called ground state. There is simply no allowed state corresponding to n=0.

    But why is there a lowest one at all?

    Why couldn’t nature have allowed one more orbit, even closer to the nucleus?

    An excellent question.

    Bohr answered where the electron can exist.

    He never really explained why those orbits exist in the first place.

    That question would eventually lead physicists to one of the deepest principles in quantum mechanics.

    We will chase that mystery in our next conversation.

  • Osmotic Pressure and a Sore Throat: Why Salt Water Gargling Works


    Ohh… I got a sore throat.
    The doctor asked me to gargle with warm salt water.

    I think when I spit the water out,
    all the bacteria will also come out with it.


    You think it is a washing process?


    Isn’t it?
    Like rinsing dirt off a plate?


    If it were that simple, plain water would be enough.

    Salt is not added for washing.

    It is added for physics.


    Physics? In gargling?


    Yes, when we add salt in water, it splits,

    \mathrm{NaCl} \rightarrow \mathrm{Na^+} + \mathrm{Cl^-}

    Particles increase.

    And when particles increase, something else increases.

    \Pi=iCRT


    Osmotic pressure…


    Yes. Osmotic pressure \Pi is the pressure created when a difference in concentration causes water to move through a semi-permeable membrane.

    It is derived using ideas from statistical mechanics and thermodynamics.

    \begin{aligned}\Pi & = \text{Osmotic pressure} \\i & = \text{van't Hoff factor (number of particles formed)} \\C & = \text{Molar concentration of the solution} \\R & = \text{Universal gas constant} \\T & = \text{Absolute temperature (in Kelvin)}\end{aligned}

    Now imagine a bacterium in your throat.

    Inside it — water.
    Outside it — your salt solution.

    Which side has higher concentration?


    Outside.


    And nature dislikes imbalance.

    Water moves from lower osmotic pressure
    to higher osmotic pressure
    through a semi-permeable membrane.

    So water leaves the bacterial cell.


    It dehydrates?


    It shrinks.

    This is like plasmolysis.

    Not because you spat it out.

    But because equilibrium demanded adjustment.


    So when I gargle, I am not washing bacteria away…


    You are disturbing their balance. It is thermodynamics.


    And the warm water?


    Look at the equation again.

    \Pi \propto T

    A little warmth,
    a little more osmotic persuasion.


    Strange.

    I thought I was just spitting bacteria out.


    No.

    You were applying

    \Pi=iCRT

    inside your throat.

  • Black Body Radiation

    Black body radiationDownload

  • Planck’s Radiation Law

    Planck radiation lawDownload

  • Compton Effect

    Compton effectDownload

  • Dual Nature of Electromagnetic Radiation

    dual nature of EM radiationDownload