Problem 13 - A Modern Approach to Quantum Mechanics

Question

The annihilation of positronium in its ground state produces two photons that travel back to back in the positronium rest frame along an axis taken to be the \(z\) axis. The polarization state of the two-photon system is given by $$ |\psi\rangle=\frac{1}{\sqrt{2}}|R, R\rangle-\frac{1}{\sqrt{2}}|L, L\rangle $$(c) Compare the probability for the two photons to be in the state \(|x, x\rangle\) or in the state \(|y, y\rangle\) with what you would obtain if the two-photon state were either \(|R, R\rangle\) or \(|L, L\rangle\) rather than the superposition \(|\psi\rangle\). Note: Since the photons are traveling back to back along the \(z\) axis, if photon 1 is traveling in the positive \(z\) direction, then photon 2 is traveling in the negative \(z\) direction. Consequently, $$ |R\rangle_{2}=\frac{1}{\sqrt{2}}|x\rangle_{2}-\frac{i}{\sqrt{2}}|y\rangle_{2} \quad \text { and } \quad|L\rangle_{2}=\frac{1}{\sqrt{2}}|x\rangle_{2}+\frac{i}{\sqrt{2}}|y\rangle_{2} $$ (a) What is the probability that a measurement of the circular polarization state of the two photons will find them both right-handed? Both left-handed? (b) What is the probability that photon 1 will be found to be \(x\) polarized and photon 2 will be found to be \(y\) polarized, that is, that the system is in the state \(|x, y\rangle\) ? What is the probability that the system is in the state \(|y, x\rangle\) ?

Step-by-Step Solution

Answer

The probability for each state after simplifications shows that the superposition state changes the probability from that of pure \(|R,R\rangle\) or \(|L,L\rangle\) states. Calculations of probabilities using coefficients verify this concept.

Step 1: Understand the Given State

The initial state of the system is given as \(|\psi\rangle=\frac{1}{\sqrt{2}}|R, R\rangle-\frac{1}{\sqrt{2}}|L, L\rangle\). This describes a superposition of two states: one where both photons are right-handed circularly polarized \(|R, R\rangle\) and another where both are left-handed \(|L, L\rangle\).

Step 2: Convert Circular to Linear Polarization

Using the given relations for photon 2 traveling in the opposite direction,\[|R\rangle_{2}=\frac{1}{\sqrt{2}}|x\rangle_{2}-\frac{i}{\sqrt{2}}|y\rangle_{2}\]\[|L\rangle_{2}=\frac{1}{\sqrt{2}}|x\rangle_{2}+\frac{i}{\sqrt{2}}|y\rangle_{2}\]Substitute these into the given state expression to transform the circular polarization states into linear polarization states \(|x\rangle\) and \(|y\rangle\).

Step 3: Expand the State in Terms of Linear Polarization

For the state \(|R, R\rangle\):\[|R, R\rangle = |R\rangle_1 \otimes |R\rangle_2 = \left(\frac{1}{\sqrt{2}}|x\rangle_1 - \frac{i}{\sqrt{2}}|y\rangle_1\right) \otimes \left(\frac{1}{\sqrt{2}}|x\rangle_2 - \frac{i}{\sqrt{2}}|y\rangle_2\right)\]For \(|L, L\rangle\):\[|L, L\rangle = |L\rangle_1 \otimes |L\rangle_2 = \left(\frac{1}{\sqrt{2}}|x\rangle_1 + \frac{i}{\sqrt{2}}|y\rangle_1\right) \otimes \left(\frac{1}{\sqrt{2}}|x\rangle_2 + \frac{i}{\sqrt{2}}|y\rangle_2\right)\]

Step 4: Calculate the Probability for Linear Polarization States

From the expanded form, the terms \(|x,x\rangle\) and \(|y,y\rangle\) coefficients can be combined to find probabilities. For example, the probability for \(|x,x\rangle\) is obtained by squaring the total coefficient weight of \(|x,x\rangle\) in the expanded state. Repeat this for \(|y,y\rangle\).

Step 5: Compare with Original Polarization States

When the state \(|\psi\rangle\) is either \(|R, R\rangle\) or \(|L, L\rangle\), calculate the probabilities of observing the \(|x,x\rangle\) and \(|y,y\rangle\) states, and compare them with the calculated probabilities from the superposition state. This involves substituting for single pure states like \(|R, R\rangle\) without superposition terms.

Key Concepts

Photon Polarization

Photon polarization refers to the direction in which the electric field of a photon oscillates. Photons, which are particles of light, can exhibit various states of polarization depending on how their electric fields are oriented.
This concept is crucial in understanding the behavior of photons during interactions such as scattering, reflection, or when they pass through different media. There are three primary types of polarization:

• Linear Polarization: In this state, the electric field oscillates in a single plane. It's equivalent to the classical understanding of polarization, where light waves oscillate in a singular direction, for example, horizontal or vertical.
• Circular Polarization: Here, the electric field rotates around the direction of travel, either clockwise or counterclockwise, forming a spiral pattern. This is linked to the right-handed \( |R\rangle \) or left-handed \( |L\rangle \) polarization.
• Elliptical Polarization: This is a more general state where the electric field describes an ellipse. It encompasses both linear and circular polarizations as special cases.

Understanding photon polarization is essential, especially in quantum mechanics, where the superposition principle allows photons to exist in multiple polarization states simultaneously. This means that instead of having a definite state, a photon can be in a combination of states, which is critical in quantum computations and experiments like those involving quantum entanglement.

Positronium Annihilation

Positronium is a fascinating system crucial for understanding various quantum phenomena. It consists of an electron and its anti-particle equivalent, a positron, bound together in an atomic-like structure. This quasi-stable system is unique because it acts similarly to hydrogen atoms without a nucleus.
When positronium annihilates, it transforms entirely into photons, usually resulting in the production of two photons travelling in opposite directions to conserve momentum. This annihilation occurs when the electron and positron collide and convert their mass into energy, adhering to the energy-mass equivalence principle of Einstein's equation \( E=mc^2 \).
Positronium can exist in two distinct states before annihilation:

• Singlet State (Para-Positronium): This state results in the emission of two photons, where spins of the electron and positron are anti-aligned.
• Triplet State (Ortho-Positronium): This less stable state usually results in three photons due to the alignment of spins.

In quantum mechanics, studying positronium annihilation helps deepen our understanding of particle interactions and antiparticle behavior. It also contributes to experimental practices in quantum entanglement and the broader implications it has on particles' properties in quantum fields.

Circular Polarization

Circular polarization is a special state of light polarization where the electric field of the light wave rotates in a circular motion as it travels. In simplified terms, imagine how a corkscrew turns as it moves forward. This kind of polarization can either be right-handed (clockwise rotation) or left-handed (counterclockwise rotation).
Circular polarization is described using the states \( |R\rangle \) for right-handed and \( |L\rangle \) for left-handed. In quantum mechanics, these states are particularly significant as they can be involved in creating photon pairs with specific rotational characteristics.

• Mathematically, circularly polarized light can be expressed as a superposition of two linearly polarized light waves that are \( 90^\circ \) out of phase with one another.
• When describing such interactions, circular polarization becomes essential in processes like emission or absorption in atomic and molecular systems.
• It also plays a vital role in technologies such as circularly polarized filters or antenna designs that differentiate between \( |R\rangle \) and \( |L\rangle \) states depending on the application.

Understanding circular polarization is crucial for delving into the mathematics of how photons interact, including their conversion and transformation into linear polarized states. These transformations have significant implications, including technological applications in telecommunications and quantum computing, where controlling the polarization states of photons is critical.