Category: Uncategorized

Schrödinger and all that Jazz



Schrödinger was born in Vienna in 1887. He was an exemplary schoolboy, displaying a startling ability in all his classes. He taught himself English and French in his spare time, and nurtured a love of classical literature. By the time he enrolled at the University of Vienna in 1906 he was focused on physics, but still took the time to learn a great deal of biology, which informed his later work – contributions that were cited as inspirational by the discoverers of DNA.

The work for which he is remembered requires some context. As with all science, an individual’s contributions to physics rarely occur in a vacuum. A host of other figures had set the stage for Schrödinger’s entrance. His seminal work began with his attempts to resolve a central mystery of the nascent quantum theory and to understand it let me take you back down the lane when Physics got a new interpretation.

After the startling discovery of the electron in 1897 by J.J.Thompson a new era of  thoughts behind the modeling of the atom began. A comprehensive model where the electron revolves around the nucleus seemed to have satisfied the  puritanical notion at that point of time, but it was still unclear among physicists, the reason for the electrons not spiraling into the nucleus. If the electron did revolve around the nucleus in fixed paths known as orbits, then by orbiting, electron should lose energy, go lower in its orbit around the nucleus where it should revolve faster and thus emit more energy. Eventually, the electron will collapse into the nucleus. But if such phenomena happened then there would be  a complete annihilation and immense amount of energy would have been emitted thus prompting in the end that no living being and so far as that is concerned, nothing would have ever appeared.

In 1900, Max Planck had discovered that the precise nature of the radiation emitted by hot objects could only be explained if the energy of the radiation came in discrete lumps that came to be known as ‘quanta’. He suggested that electromagnetic energy could only be emitted in quantized form, i.e. the energy could only be a multiple of an elementary unit i.e., E = hf, where h is Planck’s constant and f is the frequency of the radiation. The idea soon paved way for Bohr to hypothesize that negatively charged electrons revolve around a positively charged nucleus at certain fixed “quantum” distances and that each of these “spherical orbits” has a specific energy associated with it such that electron movements between orbits requires “quantum” emissions or absorption of energy. By this time, Einstein had also suggested that light can behave as a wave as well as like a particle i.e, it has dual character. Finally, enter the scene- Loius de Broglie. In 1921, he simply rearranged the already know equation between momentum and wavelength(p=h/ λ) thus giving rise to:

λ = h/mv

This raised a motion arguing that just as light which was always thought to be a wave now showed particle like behaviour then can all microscopic particles such as electrons, protons, atoms, molecules etc. also have a dual character? According to de Broglie the answer was “Yes.” He stated that any material body in motion can have wavelength but it is measurable or significant only for microscopic bodies such as electron, proton, atom or molecule.


In 1926, it was this framework that Schrödinger used to develop the ideas in his paper “Quantisation as an Eigenvalue Problem” , which contains the wave equation that bears his name. He said that if electrons really were waves then there motions should be described by wave formulae rather than Newtonian mechanics. Thus, he described a wave equation which was nothing but a mathematical distribution of a charge of an electron distributed through space, being spherically symmetric or prominent in certain directions, i.e. directed valence bonds, which give the correct values for spectral lines of the hydrogen atom. And it is to be noted that electrons won’t orbit the nucleus in the sense planet orbits around the sun, but instead exist as STANDING WAVES.

Now to understand the depth of this beautiful equation we start our journey from the fundamentals of classical mechanics where,

Total energy (E)= Kinetic Energy + Potential energy

                                                 E = ½ mv2 + V(x)                                (i)

As p=mv therefore,            E = p2/2m + V(x)                                         (ii)

Considering a standing wave given by the function:

Ψ(x,t) = A e ikx = A[cos(kx) + isin (kx)]

Hence the function for a travelling wave can be written as:

Ψ(x,t) = A e i(kx−ωt)

A travelling wave

Here the term ωt gives us the velocity of the travelling wave. We can also say that the wave number(k) is a function of the wavelength as well as momentum, k=2π/λ= p/ ħ. Now differentiating the wavefunction with respect to time we get,

∂Ψ /∂t = −i ω Ψ (keeping x constant)

And, double differentiating the wavefunction with respect to x we get,

                                       ∂2Ψ /∂x2= −k2Ψ (keeping time constant)

As already described earlier,     k=p/ ħ

Hence,                            – ħ2 (∂2Ψ /∂x2 ) = p2 Ψ

We now generalize this to the situation in which there is both a kinetic energy and a potential energy present, then according to equation (ii) we get,

EΨ = (p2/2m)Ψ + V(x) Ψ

Replacing p2 Ψ by  – ħ2 (∂2Ψ /∂x2 ), we obtain:

Eqn (a)

Now this is the time independent Schrodinger equation.

We can also write that: E= ħ ω

hence, i ħ (∂Ψ /∂t) = ħ ωψ = EΨ                          (iv)

Therefore, from equation (ii) and (iii) we get,

                             i ħ (∂ψ /∂t) = (−ħ 2/ 2m) (∂2ψ /∂x2) +V(x)Ψ

This is the final Schrodinger time dependent equation. Now to understand the equation in more details we carry out the energy state analysis of the equation for a hydrogen atom.


Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row is an example of a state which is not a stationary state. The right column illustrates why stationary states are called “stationary”.

Keeping in mind three fundamentals we can easily go around solving the equation, the three condition being,

  1. Total energy (E)= Kinetic Energy + Potential energy.
  2. The net centripetal force experience by the electron is balanced out by the columb force that helps keep the electron in the orbit around the nucleus.
  3. Finally, the angular momentum of the electrons are quantized.

In equation (i) the potential energy term are in polar coordinates,

                                   V(x)= – Ze2/4πr

the rest of the equation not being in polar coordinates thus needs to be converted. After converting and considering eqn(a) along with all the above mentioned conditions, we frame from equation that,

Energy(E)= – [e4m/8ε02r] (Z2/n2)     

Therefore,                      E= – R (Z2/n2)                                                        (v)

Where R= [e4m/8ε02r] =13.6 eV is know as the Rydberg constant and as we are considering a Hydrogen atom hence Z=1 and n=1,2,3…

So, from equation (v) it is clear that,

                                      E= – (13.6 * 1/n2)  eV

Therefore, we observe that the energy states allowed for each electron is constrained by this equation. Energy levels are governed by the integer values of n. Thus, we can only have few specific values of energy depending on the values of the quantum number(n). We also see that the energy levels are inversely proportional to the quantum numbers thus as we go further away from the nucleus the energy levels decreases.

Quantum mechanically electrons are thus considered as waves which are satisfied by the Schrodinger’s equation. Schrodinger latter also introduced the Hamiltonian operator to represent the total energy of the system

Ĥ= – ħ2 (∂2Ψ /∂x2 )

thus, turning the equation in to,

The time-independent equation can be solved analytically for a number of simple systems. The time-dependent equation is of the first order in time but of the second order with respect to the co-ordinates, therefore it is not consistent with relativity. The solutions for bound systems will give three quantum numbers, corresponding to three co-ordinates, and an approximate relativistic correction can be done by including fourth spin quantum number.

Hence this is how a person who once commented on Quantum mechanics stating, “do not like it, and I am sorry I ever had anything to do with it”  framed the Jazziest equation of all time!
 

Is everything quantum entangled?

 If the Big Bang is the creation of everything – time included, and this came from a single “source”, would it not follow that all particles in the universe are entangled with each other? And perhaps, we cannot observe this entanglement because we are “inside the system”. Are we not all connected? Questions like these have haunted physicist for almost a century now, but the answer to this was always pretty clear. To understand the depth of this question first we need to understand the concept of quantum entanglement that had bothered scientists like Neils Bohr and Albert Einstein for the entirety of their careers. Einstein was so concerned by the idea that he once famously described entanglement to be a “spooky action at a distance”.

By definition Quantum entanglement is a physical phenomenon which occurs when pairs or groups of particles are generated, interact, or share spatial proximity in ways such that the quantum state of each particle cannot be described independently of the state of the other(s), even when the particles are separated by a large distance—instead, a quantum state must be described for the system as a whole. In simpler words an entangled state is a state that is not fully separable, i.e. one that is not probability mixture of product states. This is just the quantum version of the notion of “independent random variables” in more ordinary probability theory.

For any compound system, almost all states are entangled, as the non-entangled ones vanish (measure zero). For example, any time you measure a particle with apparatus, after the measurement the apparatus indicates something about the measured system. Thus, the joint state of the compound “apparatus + particle” system is not separable because its parts are not independent of one another. Measurement produces a particular case of entanglement between the measuring and the measured.

As a macroscopic system interacts with its environment, information about it diffuses into the environment, producing entanglement between parts of the system and parts of its environment. Overall, this leakage of information is responsible for quantum decoherence and the second law of thermodynamics. But its chaotic nature means that for all practical purposes, that information is lost, and we can think of entropy increase as ‘missing information’.

However, if we just pick particles at random, chances that we can treat them as essentially independent—unentangled is overwhelming. Intuitively, one can think information about something becoming so mixed up in its environment at large that any particular microscopic parts of it and its environment are going to tell us next to nothing about each other. In other words, they’d be almost entirely independent from each other–unentangled.

By far the most particles in the visible universe aren’t quantum entangled. That’s obvious from observation, since if all electron spins would be entangled, all electrons would flip their spin at the same time, and we couldn’t observe varying statistics of electron spins in a sample, resulting in varying degrees of magnetism after the application of a  gradually changing exterior magnetic fields. Also, all (bosonic) atoms of spin 0 would form a Bose-Einstein condensate not just near absolute zero, since they would be in the same quantum state, hence in the lowest.

If we try to consider some hypothetical state of all particles, that state is almost certainly entangled. However, that does not mean that we can’t (say) treat two particles at random as very probably independent of each other–effectively unentangled.

The “single source” isn’t a single quantum state. In the early phase of the big bang temperatures were very high, allowing for a huge amount of possible quantum states within a small volume. Without warranty we may think of the initial state of the universe as a superposition of all possible quantum states, the wave functions of which collapse relative to an observer, regarding the observer as a particular quantum state.

That way we get to the many world interpretation of quantum theory and a universal wave function. Hartle-Hawking state as a pre-Planck epoch of the universe can also be considered here.