For a Deeper Understanding and Appreciation of Quantum Physics

LOWEST-ORDER RELATIVISTIC ENERGY CORRECTION OF 1-D HARMONIC OSCILLATOR

Lotis R. Racines and Edwin B. Fabillar

In quantum mechanics, relativistic correction to the energy levels of a system is used when it is introduced by a little disturbance we often recognized as $\lambda$ . Fine structure is an example of this where the splitting of spectral lines of atoms is due to its first-order relativistic corrections. Here is an example of finding the first-order relativistic corrections of a given system.

Our task is to find the lowest-order relativistic corrections to the energy levels of the one-dimensional harmonic oscillator.

Note: Our reference through all these is Jackson’s book of Quantum Mechanics

Start:

We begin by eq’n 6.52 ,

$E_{r}^' = - \frac {1}{2mc^2} [E^2 - 2 E\langle V \rangle + \langle V^2 \rangle]$

where $E = (n + \frac {1}{2}) \hbar \omega$

and $V = \frac {1}{2} m \omega ^2 x ^2$

So,

$E_{r}^' = - \frac {1}{2mc^2} [(n + \frac{1}{2})^2 \hbar^2 \omega ^2 - 2(n + \frac{1}{2}) \hbar \omega (\frac{1}{2}m \omega ^2) \langle x^2 \rangle + \frac{1}{4} m^2 \omega ^4 \langle x^4 \rangle]$

with $\langle x^2 \rangle = (n + \frac{1}{2}) \frac{\hbar}{m \omega}$

Substituting this, we get

$E_{r}^' = - \frac{1}{2mc^2} [(n + \frac{1}{2})^2 \hbar ^2 \omega ^2 - (n + \frac{1}{2})(n + \frac{1}{2}) \frac{\hbar}{m \omega} (\hbar \omega)(m \omega ^2)$

$+ \frac{1}{2} m^2 \omega^4 \langle x^4 \rangle]$

$E_{r}^' = - \frac{1}{2mc^2} [(n + \frac{1}{2})^2 \hbar^2 \omega ^2 - (n + \frac{1}{2})^2 \hbar^2 \omega ^2 + \frac{1}{4} m^2 \omega ^4 \langle x^4 \rangle]$

$E_{r}^' = - \frac{m \omega ^4}{8c^2} \langle x^4 \rangle$ (1)

We now introduce the ladder operators. That is,

$a_{+} = \sqrt {n+1}|n \rangle$

$a_{-} = \sqrt {n} |n-1 \rangle$

Using these, we could then derive $\langle x^4 \rangle$ basing on Eq’n 2.69,

$x^4 = \frac{\hbar^2}{4m^2 \omega ^2} (a_{+}^2 + a_{+}a_{-} + a_{-}a_{+} + a_{-}^2)(a_{+}^2 + a_{+}a_{-} + a_{-}a_{+} + a_{-}^2)$

$\langle x^4 \rangle = \frac{\hbar^2}{4m^2 \omega ^2} \langle m | (a_{+}^2 a_{-}^2 + a_{+}a_{-}a_{+}a_{-} + a_{+}a_{-}a_{-}a_{+} + a_{-}a_{+}a_{+}a_{-}$

$+ a_{-}a_{+}a_{-}a_{+} + a_{-}^2 a_{+}^2) | n \rangle$

Note that only equal numbers of raising and lowering operators will survive.

By eq’n 2.66, $h(\x) = h_{even} (\x) + h_{odd} (\x)$

$\langle x^4 \rangle = \frac{\hbar ^2}{4m^2 \omega ^2}{\langle n| a_{+}^2[\sqrt {n(n-1)}|n-2\rangle] + a_{+} a_{-} \langle n|n \rangle + a_{+} a_{-} \langle (n+1)|n \rangle$

$+ a_{-} a_{+} \langle n|n \rangle + a_{-} a_{+} \langle (n+1)|n \rangle + a_{-}^2 \langle \sqrt {(n+1)(n+2)}|n+2 \rangle}$

$\langle x^4 \rangle = \frac{\hbar^2}{4m^2 \omega^2} \{\langle n|[\sqrt{n(n-1)}\sqrt{n(n-1)}|n \rangle] + n \langle n|n \rangle + (n+1) \langle n|n \rangle$

$+ n \langle (n+1)|n \rangle + (n+1) \langle (n+1)|n \rangle + \sqrt{(n+1)(n+2)} \langle \sqrt {(n+1)(n+2)}|n \rangle \}$

$\langle x^4 \rangle = \frac{\hbar ^2}{4m^2 \omega ^2} [n(n-1) + n^2 + (n+1)n + n(n+1) + (n+1)^2 + (n+1)(n+2)]$

$\langle x^4 \rangle = (\frac{\hbar}{2m \omega})^2 [n^{2} - n + n^2 + n^2 + n + n^2 + n + n^2 + 2n + 1 + n^2 + 3n +2]$

$\langle x^4 \rangle = (\frac{\hbar}{2m \omega})^2 [6n^2 + 6n + 3]$

Going back to (1) to get $E_{r}^'$ ,

$E_{r}^'= - \frac{m \omega ^4}{8c^2} (\frac{\hbar ^2}{4 m^2 \omega ^2}) (3)(3n^2 + 2n + 1)$

Thus, the lowest-order relativistic correction of one-dimensional harmonic oscillator is

$E_{r}^' = - \frac{3}{32} (\frac {\omega ^2 \hbar ^2}{mc^2}) (3n^2 + 2n +1)$

Share and Enjoy: These icons link to social bookmarking sites where readers can share and discover new web pages.

Posted in Quantum Science Philippines | No Comments »

On the EPR paper “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?”

LUTCHIE DYAN S. MENDOZA

In the May 15, 1935 issue of Physical Review , Albert Einstein co-authored a paper with his two postdoctoral research associates at the Institute for Advanced Study, Boris Podolsky and Nathan Rosen. The paper, known as EPR, became a centerpiece in debates, challenging the validity of Quantum Theory.

The paper features a striking case where two quantum systems interact in such a way as to link both their spatial coordinates in a certain direction and also their linear momenta (in the same direction). As a result of this “entanglement”, determining either position or momentum for one system would fix (respectively) the position or the momentum of the other. In quantum mechanics, in the case of two physical quantities described by non- commuting operators, the knowledge of one prevents the knowledge of the other (Heisenberg Uncertainty Principle). Thus, the paper asserts that, either (1) the quantum- mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities, like position and momentum, do not commute the two quantities can not have simultaneous reality. The authors affirm that one or another of these assertions must hold, giving rise to these two premises: (1) if quantum mechanics were complete (first option failed) then the second option would hold, that is, incompatible quantities cannot have real values simultaneously but (2) that if quantum mechanics were complete, then incompatible quantities (in particular position and momentum) could indeed have simultaneous, real values. They conclude that quantum mechanics is incomplete. The conclusion certainly follows since otherwise if the theory were complete one would have a contradiction. To establish the premises, the authors discuss the idea of a complete theory, offering only a necessary condition. In order for a theory to be complete, every element of the physical reality must have a counterpart in the physical theory, further requiring the criterion: If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity. This is the famous EPR Criterion of Reality.

To realize their assertions further, the authors provided a thought experiment wherein the two quantum systems (one system is labeled Albert while the other system is named Neils) interact in such a way that conservation of relative position and momentum hold following their interaction. The paper constructs an explicit wave function for the combined system that satisfies both conservation principles. The critical point in the paper centers on the two assumptions made by the authors, namely, separability and locality. The first assumption states that at the time when measurements will be performed on Albert’s system there is some reality that pertains to Niels’ system alone. In effect, they assume that Niels’ system maintains its separate identity even though it is correlated with Albert’s. The second assumption supposes that no real change can take place in Niels’ system as a consequence of a measurement made on Albert’s system. Locality implies that the prediction of the position of Niels’ system does not involve any change in the reality of Niels’ system. Since the prediction does not disturb Neils’ system, all the pieces are in place to apply the Criterion of Reality. Hence, the authors concluded that Niels’ system can have real values or elements of reality for both position and momentum simultaneously. The negation of the first premise leads to the negation of the only alternative.

Following the result of the thought experiment, separability, locality as well as the application of the Criterion of Reality, EPR concludes that quantum- mechanical description of a physical reality given by the wave functions is not complete.

Reference:

A. EINSTEIN, N. ROSEN and B. PODOLSKY, Phys. Rev. 47, 777 (1935).

Share and Enjoy: These icons link to social bookmarking sites where readers can share and discover new web pages.

Posted in Quantum Science Philippines | No Comments »

Commutation of Spin, Angular and Spin-Orbital Momentum

Marichu T. Miscala

In quantum mechanics, the presence of spin-orbit coupling gives rise to the Hamiltonian that will no longer commute with $\vec{L}$ , and $\vec{S}$ , so the spin and orbital momenta are not separately conserved.

In order to understand this concept better, a commutation problem for orbital angular momentum $\vec{L}$ , spin $\vec{S}$ , and spin-orbital momentum $\vec{J}$ is presented here.

Consider the fundamental commutation relations for angular momentum. The individual components of the spin $\vec{S}$ do not commute with each other. That is,

$[S_x,S_y] = i\hbar S_z$ ; $[S_y,S_z] = i\hbar S_x$ ; $[S_z,S_x] = i\hbar S_y$ ;

The commutation relation for the ‘instrinsic’ angular momentum $\vec{S}$ is much like a ‘carbon copy’ to that of the ‘extrinsic’ angular momentum $\vec{L}$

$[L_x,L_y] = i\hbar L_z$ ; $[L_y,L_z] = i\hbar L_x$ ; $[L_z,L_x] = i\hbar L_y$ ;

But the spin-obit Hamiltonian does commute with $L^2$ , $S^2$ and the total angular momentum $\vec{J}$ which is

$\vec{J} = \vec{L} + \vec{S}$

From the given fundamental commutation relations above, we then now seek the commutators of the following commutation relations:

(a.) $[\vec{L}.\vec{S},\vec{L}]$ (b.) $[\vec{L}.\vec{S},\vec{S}]$ (c.) $[\vec{L}.\vec{S},J]$

(d.) $[\vec{L}.\vec{S},L^2]$ (e.) $[\vec{L}.\vec{S},S^2]$ (f.) $[\vec{L}.\vec{S},J^2]$

As a hint here, $\vec{L}$ and $\vec{S}$ satisfy the fundamental commutation relations for angular momentum, but $\vec{L}$ and $\vec{S}$ commute with each other.

(a.) $[\vec{L} . \vec{S} , L_x] = [L_x S_x + L_y S_y + L_z S_z , L_x]$

$= S_x [L_x, L_x] + S_y [L_y, L_x] + S_z [L_z, L_x]$
$= S_x (0) + S_y (-i \hbar L_z) + S_z (i \hbar L_y)$
$= i \hbar (L_y S_z - L_z S_y) = i \hbar (\vec{L} \times \vec{S})_x$

same goes for the other two-components, so

$[\vec{L}.\vec{S}, \vec{L}] = i \hbar (\vec{L} \times \vec {S})$

(b.) $[\vec{L}.\vec{S},\vec{S}]$ is identical only with $\vec{L} \Longleftrightarrow \vec{S}$

$[\vec{L}.\vec{S},\vec{S}] = i \hbar (\vec{S} \times \vec{L})$

(c.) $[\vec{L}.\vec{S},\vec{J}] = [\vec{L}.\vec{S},\vec{L}] + [\vec{L}.\vec{S},\vec{S}] = i \hbar (\vec{L} \times \vec{S} + \vec{S} \times \vec{L}) = 0$

(d.) $L^2$ commutes with all the components of $\vec{L}$ (and $\vec{S}$ ), so

$[\vec{L} . \vec{S} , L^2] = 0$

(e.) Likewise,

$[\vec{L} . \vec{S} , S^2] = 0$

(f.) $[\vec{L} . \vec{S} , J^2] = [\vec{L} . \vec{S} , L^2] + [\vec{L} . \vec{S} , S^2] + 2[\vec{L} . \vec{S} , \vec{L} . \vec{S}]$

where $J^2 = (\vec{L} + \vec{S}) . (\vec{L} + \vec{S}) = L^2 + S^2 + 2 \vec{L} . \vec{S}$

The first, second and third terms vanish from the results of (d.) and (e.). Thus,

$[\vec{L} . \vec{S} , J^2] = 0$

This means that the quantities $L^2$ , $S^2$ and $\vec{J}$ are conserved. That is, the eigenstates of $L_z$ and $S_z$ are not “good” states to use in perturbation theory, but the eigenstates of $L^2$ , $S^2$ and $J_z$ are.

The problem presented here is based on the problem 6.16 from D.J. Griffiths “Introduction to Quantum Mechanics”.

About the author: Marichu T. Miscala is currently taking up her masters in Mindanao State University – Iligan Institute of Technology. She is most interested in pursuing a career in advanced research.

Share and Enjoy: These icons link to social bookmarking sites where readers can share and discover new web pages.

Posted in Quantum Science Philippines | No Comments »

Quantum Mechanics Violating Bell’s Inequality

Lotis R. Racines

Einstein never liked quantum mechanics. He didn’t like the idea that the momentum of a particle, if it’s position was known, was completely unknowable: that is, random [1]. He even said that: “God does not play dice with the universe.” which was referred to Copenhagen Interpretation of quantum mechanics that there exists no objective physical reality other than that which is revealed through measurement and observation.

Einstein wasn’t the only one who didn’t like Quantum Mechanics. In 1935 he got two other physicists, Boris Podolsky and Nathan Rosen, and wrote a famous paper entitled Can Quantum-Mechanical Description of Physical Reality be Considered Complete? We now refer to it as simply the EPR Paradox. EPR Paradox occurs when phenomenon appears disobeying local realism, EPR’s assumption. With this, EPR concluded that since there were “real” properties of the world not even definable in quantum theory, then quantum theory is “incomplete”.

Aside of Einstein, Podolsky and Rosen, there were already several physicists trying to convey quantum mechanics. One of these physicists was David Joseph Bohm, a British quantum physicist who made significant contributions in the fields of theoretical physics, philosophy, neuropsychology and to the Manhattan Project. He published his first book, Quantum Theory, on 1951 but was not satisfied with some he had written in that book. So he began to develop his own approach, Bohm interpretation, a non-local hidden variable deterministic theory whose predictions agree perfectly with the nondeterministic quantum theory. In 1959, Bohm and his student, Yakir Aharonov, discovered the Aharonov-Bohm effect, which states that a quantum mechanical phenomenon in which an electrically charged particle shows a measurable interaction with an electromagnetic field despite being confined to a region in which both the magnetic field B and electric field E are zero. (In fact this effect was predicted first by Werner Ehrenberg and Raymond Siday in 1949.) He then claimed that either we abandon the locality principle or we are forced to accept the realization that the electromagnetic potential offers a more complete description of electromagnetism than the electric and magnetic fields can [2]. His work and the EPR argument became the major motivation of John Bell deriving the Bell’s theorem.

In 1964, after a year’s leave from CERN that he spent at Stanford University, the University of Wisconsin-Madison and Brandeis University, he wrote a paper entitled “On the Einstein-Podolsky-Rosen Paradox“. Applying the EPR’s assumption, a mathematical relation that was expressed by an inequality shown below, was derived concerning outcome of some measurements of microscopic particles.

$1 + P(\vec{b},\vec{c}) \ge | P(\vec{a},\vec{b}) - P(\vec{a},\vec{c})|$

With this, he showed that carrying forward EPR’s analysispermits one to derive the Bell’s inequality. This inequality conflicts with the predictions of quantum theory. That is, Bell’s inequality shows that there are limits that apply to local hidden variable modes of quantum systems, and that quantum mechanics predicts that these limits will be exceeded by measurements performed on entangled pairs of particles.

Now, there are several yet different responses [3] regarding violation of Bell’s inequality.

to simply assume that quantum mechanics is wrong. However, experiments undergone have supported quantum mechanics!
to abandon the notion of hidden variables and to argue that the wave function does not contain any information about the outcome of the measurement of the values in the particles which leads to the Copenhagen interpretation of quantum mechanics.
to give up locality in favor of the non-locality principle which leads to Bohm’s interpretation of quantum mechanics. However, this type of interpretation is regarded as inelegant, since it requires all particles in the universe to be able to instantaneously exchange information with all other particles in the universe!
to assume counterfactual definiteness. However, in the Everett many-worlds interpretation, the assumption of counterfactual definiteness is abandoned because this interpretation assumes that the universe branches into many different observers each which measures a different observation!

After Bell’s article appear many experiments from all over the world to test Bell’s theorem. But the most interesting experiment was carried out by a physicist at the University of Geneva, Switzerland, Nicolas Gisin in 1997. He was able to measure entangled particles (emanating from a single particle to two particles) about 10 kilometers apart through a detector. This could mean that particles no matter how distant they maybe, could somehow communicate with each other and somehow affect each others’ measurement.

Up to these days, Bell’s inequality with EPR’s assumed principles is still being debated. However, most of the experiments we’re to violate Bell’s Inequality which implies that locality should not be generally assumed, since there are instances that an action of a particle could affect the behavior of the other particle though remote; hence, pushes quantum theory as a better candidate to theoretical and applied physics.

This was based from John Bell’s article in 1964 entitled, “On the Einstein-Podolsky-Rosen Paradox” [4] which basically shows that the principle (“local realism”) assumed by Einstein and colleagues contradicts to the statistical predictions of quantum mechanics.

Reference:

[1] http://library.thinkquest.org/C008537/cool/bellsinequality/bellsinequality.html

[2] http://en.wikipedia.org/wiki/David_Bohm

[3] http://www.starrepublic.org/encyclopedia/wikipedia/b/be/bell_s_theorem.html

[4] J. S. Bell, “On the Einstein Podolsky Rosen Paradox”, Physics Vol. 1, No. 3, 195-200, 1964

**************************************************************************************************

If there’s no separation of reality from the observer, could it be that the universe only exists because we are conscious of it? Or perhaps we only exist because someone or something is conscious of us? This is where physics wonderfully melds with philosophy and religion.

**************************************************************************************************

A little about the Author:

Lotis R. Racines is an MS Physics Student of Mindanao State University – Iligan Institute of Technology, Iligan City. Hopefully, she’ll take up other fields of Science after her Master’s Degree in MSU-IIT.

Share and Enjoy: These icons link to social bookmarking sites where readers can share and discover new web pages.

Posted in Quantum Science Philippines | 1 Comment »

Three Dimensional Virial Theorem for the Hydrogen Atom

Catherine Therese J. Quiñones

The virial theorem is a general theorem relating the potential energy (V) and the kinetic energy (KE) in a bound system. A simple physical example is a small object orbiting around another object bound by a force. The virial theorem states that the potential energy is twice the kinetic energy thus,

$<V>=2<T>$ (1)

Here, we will derive the expectation value of $1/r$ in the unperturbed state of the of a hydrogen atom. We can use the virial theorem to easily solve the expectation value since the system can be considered a bound system with the electron orbiting around the proton bound by the Coulombic force. Hence, the potential energy is

$V=-\frac{e^2}{4\pi\epsilon_0}\frac{1}{r}$ (2)

where $e$ is the charge of the electron and the proton, $r$ represents the separation distance between the two charges and $\epsilon_0$ is the permittivity of free space . The negative sign indicates that the force is attractive. The total energy is given by

$E_n = - \Big[\frac{m}{2\hbar^2}\Big(\frac{e^2}{4\pi\epsilon_0}\Big)^2\Big]\frac{1}{n^2}$ (3)

where $m$ is the mass of the particle, $\hbar$ is Planck’s constant over $2\pi$ and $n= 0,1,2,3,..$ which indicates the quantization of the energy level. The solution is very straight forward. All we need is to plug in eqn (2) and (3) to eqn (1). Hence,

$<V>= 2E_n$ (4)

$-\frac{e^2}{4\pi\epsilon_0}<\frac{1}{r}> = -2\Big[ \frac{m}{2\hbar^2}\Big(\frac{e^2}{4\pi\epsilon_0}\Big)^2\Big] \frac{1}{n^2}$ (5)

$<\frac{1}{r}>=\Big(\frac{me^2}{4\pi\epsilon_0\hbar^2}\Big)\frac{1}{n^2}$ (6)

Note that the term inside the parenthesis is just $\frac{1}{a_0}$ , where $a_0$ is the Bohr radius . Hence we can write the expectation value of $1/r$ as,