CHAPTER 8 : Time: Reality We Live or Illusion We Believe?

 

    What exactly is time? Is it just an imaginary thing or does it exists in real life? Is time just an imaginary part of the complex equation? We must have heard this thing that time is delusional maybe in you tube videos and some ancient scriptures. But it is just the conclusion made like the water surface but unexplored deep inside as light rays can't reach so deep. The light rays is just like a human way of thinking which is incapable to go beyond some depth of questioning ourselves. This chapter will not be philosophical, we will be going to discuss how the most accurate time is measured.

    The atomic clock is the most accurate measurement of time in the modern times. As mentioned in the previous chapter that the protons and neutrons also contains the magnetic field but it is very small as compared to that in electrons, so in the SG apparatus that interaction between of the nucleus spin (proton + neutron spin) with the electron spin is ignored. For measuring time with atomic clock, this interaction  between nucleus spin and electron spin is used. So like the silver atom was used in the SG experiment for the reasons mentioned in the previous chapter, for measuring time with atomic clock, we will use cesium - 133 atom. Again the question arises why choose only cesium-133 atom?

    Cesium atom is the only atom in the universe explored till now which has only one naturally occurring isotope: Cs-133. It basically means that every cesium atom in the universe is identical in nuclear structure avoiding any confusion from multiple isotopes. This ensures consistency and universality for time measurement. What exactly is this number 133 and how does it came? The cesium atom contains: 55 protons, 78 neutrons so its total nuclear mass is 133 units ( in atomic mass units) and 133 - 78 = 55 electrons. 

    As discussed and proved in previous chapter, spin is the quantized state. Like silver atom in SG experiment has spin quantum number as S= +1/2 and/or -1/2, the cesium atom also has the same spin quantum number. It has nucleus spin of I = 7/2. The electron spin value came from the SG experiment and the value of nucleus spin came from other experiments of Hyperfine spectroscopy, Nuclear Magnetic Resonance (NMR). The NMR experiment will be explained in the other chapter. The hyperfine splitting (which will be explained in this chapter) is responsible for atomic clock measurement and the value of I. What is hyperfine splitting?

    As there is interaction between the nucleus spin and electron spin due to magnetic interaction. There is always alignment and anti-alignment between the nucleus and electron spin. Depending on whether they are pointing mostly in the same (aligned) direction or the opposite (anti-aligned) direction, the total energy of the atom changes slightly. This small difference between the two arrangements is called hyperfine-splitting. So, in cesium-133 this hyperfine splitting creates two possible energy states in the ground level: one with total spin F=3 (anti-aligned) and one with total spin F=4 (aligned). The gap or jumping between them corresponds to the clock frequency. What exactly is this Energy and what exactly is F here and how did these values came?

    F refers to the total angular momentum quantum number of the atom in its ground state (electron + orbital + nucleus). The values came from:
F = I+J     &                                              .... a)
J = L+S                                                     .... b)
here J = total angular momentum only due to electrons and orbit 
L = orbital angular momentum
In the equation b) substitute L=0, as the only one valence electron present in cesium is in s orbital and also substitute S = + 1/2 and/or -1/2 in equation b and the result will be J = +1/2 and/or -1/2. Accordingly, F=3 and 4 came by taking J as -1/2 and +1/2 respectively and taking I = 7/2 in the equation a). To calculate the energy for F=4 and F=3, we use the following formula:

i) Energy convention

$$E_F=\frac{A}{2}[F(F+1)-I(I+1)-J(J+1)] \; \; \; ....\; c)$$

• $A$ has units of energy.

• $A$ is called the magnetic dipole hyperfine structure constant.

• It tells us how strongly the magnetic field from the nucleus interacts with the magnetic field from the electron’s spin and orbital motion.

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ii) Frequency convention

$$\nu_F = \frac{A_\nu}{2} \, \big[ F(F+1) - I(I+1) - J(J+1) \big]$$

• $A_\nu$ has units of Hz.

• Relation to energy:

  $$E_F=h{\nu }_{\mathrm{F}}$$

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Cesium-133 hyperfine splitting

For the cesium-133 ground state, the measured splitting frequency difference between $F=\mathrm{<span\; class="base"><span\; class="strut"></span><span\; class="mord">4\; and\; F=3\; is: </span></span>}$

$$\mathrm{\Delta }\nu =\mathrm{9,192,631,770}\text{ Hz}$$

For an $S_{1/2}$ state (with $J=\tfrac{1}{2}$ and $I=\tfrac{7}{2}$), the two hyperfine levels are separated by:

$$\Delta \nu = 4 A_\nu \quad \Rightarrow \quad A_\nu = \frac{\Delta \nu}{4}$$

The exact value of $\Delta \nu$ is measured experimentally (using the Cesium Atomic Beam Clock or the Cesium Atomic Fountain Clock) which will be discussed in the later chapter.
After substituting the value of $\Delta \nu$, we can determine $A_\nu$, and from there calculate $\nu_F$

If you want $A$ in energy units:

$$A=h{A}_{\mathrm{\nu }}$$

and then you can compute $E_F$

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Symbols

• $E_F$: energy of a given hyperfine level.

• $\nu_F$: frequency equivalent of that energy level (via $E_F = h \nu_F$).

• $\nu$: general frequency symbol, e.g. transition frequency or splitting.

• $\Delta \nu$: difference in frequency between two levels.
  
  
  

    So, from equation c), it will be concluded that when the electron's spin and nucleus spin direction is same (aligned), the atom is in higher-energy state and when they point in the opposite direction (anti-aligned), the atom is in lower-energy state. Microwaves at exactly right frequency (9,192,631,770 Hz) can flip the system back and forth between these two states. So the frequency here basically is the number of flips done by atoms between these two alignments in one second i.e. 9,192,631,770 flips happens in one second. When microwaves of this frequency hit cesium atoms, the atoms switch (flips) back and forth between the two alignments.

    So, from the above experiment it proves that time is just calculated based on the natural behavior of atoms. Here, is the question for the readers, as every object is made up of atoms, so what happens at the quantum level to the atom's flipping behavior when any object's life span expires?( for e.g, when human dies). Share the answer if you have, through my social media links, comments.

    

    

CHAPTER 7: The Revolutionary Stern-Gerlach Experiment

    In Stern-Gerlach Experiment (we will call it SG experiment), Silver (Ag) atoms are heated in an oven. The oven contains small holes through which some of the silver atoms will escape. The fig 7.1 contains the visual representation of the SG experiment as follows:

                                                                    Fig 7.1 The SG experiment
                              


    There must be questions raised why choose Silver atom not any other? But first before getting answer to that question one should know what exactly is spin. Spin in quantum mechanics is not something we physically describe things. Spin is just the two state of any electron which is in its purest state. Purest state means here that the probability of electron to be in a specific position is 100%. The atom can be influenced by various experiments to be in its purest state. The SG experiment is one of the experiment in understanding electron's spin behavior. Spin has two states as +1/2 and -1/2 regarded as up and down positions respectively. Spin states are not fixed meaning it is not already in either of the two positions but they just move to one of the states quickly after we start observing one of the electron spins. Its called quantum entanglement in which one state influences other very quickly. This superposition of electrons and quantum entanglement idea is used in supercomputer for sharing qubits (0 or 1) information very quickly through various algorithms. The spin in an atom is from its nucleus which is a sum of spins from protons and nucleus. The other spin is from electrons. This spin creates a tiny magnetic field and magnetic moment which is going to be very important in the SG experiment.

    So, coming back to the first question raised of choosing silver atom. The silver atom is made up of nucleus and total 47 electrons where 46 out of 47 electrons can be visualized as forming a spherically symmetrical electron cloud as shown in the fig 7.1. According to Pauli's exclusion principle, in one orbital, two electrons cannot exist with same spin state. According to Pauli's exclusion principle and the following filling of orbital configuration of electrons in silver atom: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s¹ 4d¹⁰, here we see that the 5s orbital has one unpaired electron which is responsible for only spin-intrinsic angular momentum and not the orbital angular momentum because its zero for s-orbitals. The single 47th electron mass which is attached to the nucleus is very very small (~2*10^5) times smaller than the nucleus mass. So, according to the relation: μ (magnetic moment) ∝ (q/m)×(angular momentum), the magnetic moment is very very small from nucleus and the almost all the magnetic moment is just from the 47th single electron. There will not be any other external disturbance to influence the electron from not going to its spin states which is also called as quantized state because it is fixed and so μ (magnetic moment of the atom) ∝  S (electron spin). The effect of nucleus spin in electron spin is used in atomic clock. Here, in this experiment we just ignored the effect of nucleus spin. So, overall this is the reason for choosing silver atom here in the experiment.

    The z component of force experienced by the atom is : Fz​=∂​(μ​⋅B)/ ∂z = μz​ ∂Bz/∂z. In fig 7.1, μz​ > 0 ( Sz < 0) then atom experiences upward force, while μz​ < 0 ( Sz > 0) atom experiences downward force. The electrons negative changes sign inverses the direction and signs of μz​ and Sz. The beam is expected to split according to the values of μz​. The SG experiment measures the z component of μ or z component of S up to a proportionality factor.​​

    The atoms in the oven are randomly oriented; there is no preferred direction for the orientation of μ. If the electrons were like a classical spinning object, we would expect all values of μz​ to be realized between | μ | and | -μ |. This would lead us to expect a continuous bundle of beams coming out of the SG experiment as in fig 7.1 to be spread more or less evenly over the expected range. In the SG experiment, the electrons are passed through the inhomogeneous magnetic field which is essential for getting the observational result of spin. If we use homogeneous magnetic field then the electrons would just rotate and there will be no net force on electron spin due to homogeneous magnetic field and we will not be able to see the deflection in upward and downward direction of electrons due to spin. The inhomogeneous creates a varying strength magnetic field which forces the electrons to go into upward and downward direction according to its spin. The inhomogeneous magnetic field creates this net force on the electron which is essential because it converts the abstract quantum property (spin) into a spatial separation of the atom beam, which can be observed directly on the screen.

    But the actual experiment showed that there are only 2 spots observed corresponding to one "up" and one "down" orientation. The SG apparatus splits the original silver beam from the oven into 2 distinct components (space quantized). The values of Sz are: Sz​=±ℏ/2 where​ ℏ is Planck's constant = 1.0546 * 10^-27 erg-s = 6.5822 * 10^-16 eV-s. 

    In this experiment, we just setup one single SG apparatus which just quantizes the spin in only one specific direction ( z direction). In the next chapter, we are going to see what happens if the spin is quantized in multiple directions ( x, y, z) by setting multiple SG apparatus in parallel in multiple directions and what will be the classical expectation and experimental observation from this experiment.