EPR’s Paradox Exemplified: Bohm’s Spin Experiment

After my post on Einstein, Podolski and Rosen’s paradox, a few of you guys asked me about how is it possible for both the position and momentum of a particle to be elements of reality at the same time. In particular, I think this confusion may come from the choice of example I used (Eq. (3)). I decided to prepare this addendum to present another example of how entangled particles can apparently allow us to determine the values of two non-commuting operators. Thus, I want to discuss Bohm’s version of the EPR paradox. (Also, this article will be very useful for understanding Bell’s Inequalities).

Before presenting Bohm’s experiment I want you to remember that EPR’s criterion of reality establishes that in order for a physical quantity to have real values we should be able to predict with certainty (i. e. with probability equal to unity) the value of said quantities without the need of measuring or disturbing the system.

Bohm’s Experiment

In the years that followed  the EPR paper, their thought experiment continued to puzzle physicist, but it remained just that, a though experiment. It wasn’t until fifteen years after the paper was published that David Bohm proposed an experiment that could exemplify the paradox using the spins of a pair of particles.

Think of the following situation: a particle with zero spin decays into two particles (A and B), each with 1/2-spin. Due to the fact that spin angular momentum must be conserved during the decay, if initially the total spin angular momentum was zero, then after the decaying process it must still be zero. Therefore, particles A and B have opposite spin.

Bohm

Take as an example the dissociation of an exited hydrogen molecule into two hydrogen atoms. If the decaying mechanism does not change total angular momentum, then the spins on the hydrogen atoms will be anti-correlated. Whenever a measurement of the spin of A is found to be positive with respect of the z-axis (we shall note this state as |\!\uparrow\rangle), then, we could infer that the spin of the B particle must be negative (|\!\downarrow\rangle), and this is true independent of the distance that separates the particles. The spin of these particles are then entangled.

quantum_entanglement

Now, we can´t predict which spin will be positive (or negative) with respect of the z-axis, so the state that describes the spins of the particles could be for instance the spin singlet state

|\psi\rangle=\frac{|\downarrow\uparrow\rangle-|\uparrow\downarrow\rangle}{\sqrt{2}}.       (1)

We have a probability of 50% for the spin of particle A to be positive (and the spin of B negative) and a probability of 50% of it being the other way around.

So far we have assumed that we are performing a measurement along the z-axis, but measurements are not restricted to this particular election, we could measure for instance the spin of particle A along the \mathbf{\check{a}}-axis and the spin of B along the \mathbf{\check{b}}-axis. Let’s see what happens if we decide to measure the spin along the x-axis: \mathbf{\check{a}}=\mathbf{\check{b}}=\check{x}. For 1/2-spins, the spin operator S_{A/B}^{x} can be represented by the 2×2 hermitian matrix

S_{A/B}^x=\frac{1}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.       (2)

By performing a change of basis we can rewrite the state |\psi\rangle in terms of the eigenstates of (2)

|u\rangle=\frac{1}{\sqrt{2}}(|\!\downarrow\rangle + |\!\uparrow\rangle), \qquad |v\rangle=\frac{1}{\sqrt{2}}(|\!\downarrow\rangle - |\!\uparrow\rangle).       (3)

Using Eq. (3), we can rewrite the state (1) as

|\psi\rangle=\frac{1}{\sqrt{2}}(|vu\rangle-|uv\rangle).

Just like before, by choosing to measure the spin of A along the x-axis we can determine it’s value and infer the value of the spin of particle B without the need to measure it (and vice versa). Furthermore, it turns out that this is the case independent of the election of the axis we choose to measure! (Provided that \mathbf{\check{a}}=\mathbf{\check{b}}=\mathbf{\check{v}}).

This is exactly the same situation as the one we discussed in this article, a simple choice of the axis along which to measure the spin A allow us to establish the value of the spin of B along this same axis without the need to measure it. And this is also the case (as we already saw) for physical properties described by non-commuting operators (S^x and S^z do not commute).   Thus, according to EPR’s criterion of reality, these physical properties are elements of reality.

This apparent paradox was finally solved by an experiment proposed by physicist J. S. Bell. If you are interestes so far, I recommend you read my article to find out how this all turns out.


 All text copyright © Marco Vinicio Sebastian Cerezo de la Roca.

Creative Commons License
EPR’s Paradox Exemplified: Bohm’s Spin Experiment by Marco Vinicio Sebastian Cerezo de la Roca is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Advertisements

About marcocerezo

My name is Marco Cerezo, I have a degree in physics from the Universidad Nacional de la Plata (UNLP) in Buenos Aires, Argentina, and I'm a Ph.D student. Although I study in Argentina, I was born and raised in Guatemala, so I consider myself "chapín" at heart. From an early age I discovered how exciting, interesting and fun physics can be, so I decided to devote myself to study this passionate field. This blog is dedicated to all those who want to dig a little into the mysteries and beauty of physics. And in particular, to all those who are interested in studying Quantum Computation and Quantum Information. ---------------------------------------------------------------------------------------------------------- Mi nombre es Marco Cerezo, Licenciado en Física egresado de la Universidad Nacional de la Plata (UNLP) en Buenos Aires, Argentina. Actualmente estoy haciendo mi Doctorado en física. Aunque estudio en Argentina, nací y crecí en Guatemala por lo que me considero chapín de corazón. Desde muy temprana edad descubrí lo apasionante, interesante y divertida que es la física por lo que decidí dedicarme a su estudio. Este blog es está dedicado a todos aquellos que quieren indagar un poco en los misterios y la belleza de la física; y en particular, a aquellos que quieran aprender sobre Computación e Información Cuántica.
This entry was posted in Quantum Information and Quantum Computation: General Discussions and tagged , , , , . Bookmark the permalink.

2 Responses to EPR’s Paradox Exemplified: Bohm’s Spin Experiment

  1. Pingback: Entanglement (I): how it all began, the EPR Paradox. | Entangled Physics: Quantum Information & Quantum Computation

  2. Pingback: Entanglement (II): Non-locality, Hidden Variables and Bell’s Inequalities. | Entangled Physics: Quantum Information & Quantum Computation

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s