Is Quantum Superposition Real

1. The Certainty of Quantum State “Unknownness” Before Measurement

• In standard quantum mechanics, the state of a quantum system prior to measurement is not “unknown” in the classical sense, but rather it is described by a superposition of possible outcomes. The state is fully defined by the wavefunction $|\psi ⟩$, which encodes the probabilities of all possible measurement outcomes.
• The Born rule provides the connection between the wavefunction and measurement outcomes: the probability of observing a certain state is given by the square of the amplitude of the wavefunction for that state, P=∣⟨ϕ∣ψ⟩∣2P = |\langle \phi | \psi \rangle|^2P=∣⟨ϕ∣ψ⟩∣2, where $|\varphi ⟩$ is the state being measured.

Key Idea:

• Before measurement, the system is not in any definite state with respect to the observable being measured (e.g., spin along a specific axis in the SGA). Instead, it exists in a superposition of states, and the act of measurement causes the collapse of the wavefunction into one of the eigenstates of the measured observable.

2. How the Stern-Gerlach Apparatus (SGA) Works

The Stern-Gerlach apparatus measures the spin of particles (like electrons) along a specific axis, typically denoted as $z$-axis. Here’s how it works:

Basic Mechanism:

1. Magnetic Field Gradient: The SGA uses a non-uniform magnetic field that interacts with the magnetic moment of the particle. The spin of the particle is associated with its magnetic moment.
   • If the particle’s spin is aligned up ($+\hslash /2$) with the field gradient, the magnetic moment causes the particle to deflect in one direction.
   • If the spin is aligned down ($-\hslash /2$), the particle deflects in the opposite direction.
2. Spatial Separation: The magnetic field gradient splits the particles into two distinct paths based on their spin states (e.g., $+\hslash /2$ and $-\hslash /2$).
3. Detection: Detectors placed along the paths register the particle’s presence, effectively “measuring” its spin along the chosen axis.

Important Feature:

• The SGA is directional: it measures spin along the axis defined by the orientation of the magnetic field gradient. For example, an SGA oriented along the $z$-axis measures spin along $z$, while one oriented along the $x$-axis measures spin along $x$.

3. Does the SGA “Force” Electrons into a Particular Spin State?

• The SGA doesn’t “pre-determine” the spin of the particle. Rather, it measures the spin, causing the quantum system to collapse into one of the eigenstates of the spin operator along the chosen axis. This is a direct consequence of the measurement postulate in quantum mechanics.
• Before entering the SGA, the particle might be in a superposition of spin-up and spin-down states along $z$:
  $$|\psi ⟩=a|+⟩+b|-⟩,$$

  where $a$ and $b$ are complex coefficients.

• After the measurement, the wavefunction collapses into either $|+⟩$ or $|-⟩$ with probabilities ∣a∣2|a|^2∣a∣2 and ∣b∣2|b|^2∣b∣2, respectively.

Key Points:

• The SGA does not “force” the spin to take on a specific value arbitrarily. Instead, it selects from the existing superposition of states based on the probabilities encoded in the wavefunction.
• Once the spin is measured along one axis, subsequent measurements along the same axis will always yield the same result because the system is now in a definite eigenstate of that observable.

4. Entangled Electron-Positron Pairs and the SGA

When dealing with entangled pairs, such as an electron and positron, the situation becomes more intricate due to nonlocal correlations between the particles.

Entanglement and Measurement:

• Suppose an electron and positron are entangled in their spin states, such that the total spin of the system is conserved (e.g., the singlet state):
  ∣ψ⟩=12(∣+⟩e∣−⟩p−∣−⟩e∣+⟩p),|\psi\rangle = \frac{1}{\sqrt{2}} \Big(|+\rangle_e |-\rangle_p – |-\rangle_e |+\rangle_p\Big),∣ψ⟩=2​1​(∣+⟩e​∣−⟩p​−∣−⟩e​∣+⟩p​),

  where ∣+⟩e|+\rangle_e∣+⟩e​ and ∣−⟩e|-\rangle_e∣−⟩e​ represent the electron’s spin, and ∣+⟩p|+\rangle_p∣+⟩p​ and ∣−⟩p|-\rangle_p∣−⟩p​ represent the positron’s spin.

• If the positron’s spin is measured along the z-axis and found to be $+\hslash /2$, the electron’s spin immediately collapses to $-\hslash /2$ along the same axis, ensuring conservation of total spin.

Role of the SGA:

• The SGA can measure either the electron’s or positron’s spin along a chosen axis. The measurement outcome is correlated with the entangled partner due to the entangled state.
• If the SGA measures the electron’s spin, it “forces” the electron into a definite spin state along the axis of measurement, which also determines the positron’s spin state nonlocally.

5. Problem of Selective Detection

You raise an interesting concern: if the SGA detects only particles with certain spin orientations, does that bias the results?

• Selective Detection in SGA: The SGA splits particles into separate paths based on their spin. If only one path is detected (e.g., spin-up particles), then the measurement apparatus inherently introduces a filtering bias.
• Entanglement Considerations: In the case of entangled particles, measuring one particle’s spin along any axis automatically determines the spin of the other particle along the same axis. The SGA doesn’t “create” the correlation—it simply reveals the pre-existing entangled relationship.

6. Summary

1. State Unknownness: Before measurement, the spin of the electron is described by a superposition of states. The Born rule governs the probabilities of outcomes, and the state collapses upon measurement.
2. SGA Functionality: The SGA measures spin along a specific axis by spatially separating particles based on their spin states. It does not “force” the electron into a state arbitrarily but instead measures and collapses the quantum state.
3. Entangled Pairs: For entangled electron-positron pairs, the spin of one particle is correlated with the spin of the other. Measuring one particle with an SGA determines the state of its entangled partner.
4. Selective Detection: If the SGA detects only one spin state, it introduces a bias in the measurement. However, this does not invalidate the underlying quantum correlations predicted by entanglement.

The Stern-Gerlach apparatus is a powerful tool for exploring quantum measurements, but its role must be understood within the broader framework of quantum mechanics and entanglement.

1. The Role of the SGA in Forcing a Spin State

• The SGA “forces” the spin state of every particle it measures into one of two eigenstates of the spin operator along the apparatus’s chosen measurement axis (e.g., $+\hslash /2$ or $-\hslash /2$ along the $z$-axis).
• This process is what quantum mechanics refers to as wavefunction collapse. Prior to measurement, the particle’s spin is in a superposition of possible states. When the particle enters the SGA, the measurement collapses the quantum state, and the particle is deflected into one of two paths corresponding to spin-up or spin-down.

2. Single Pair Measurement: Ambiguity

If you measure just one electron-positron pair, the result (e.g., electron spin-up and positron spin-down) alone does not tell you whether:

1. The particles were originally in those spin states prior to measurement.
2. The SGA forced the spin states into these values during the measurement process.

This ambiguity arises because:

• The quantum state prior to measurement is not directly observable in quantum mechanics.
• Measurement only gives a definite outcome (up or down) for each particle, but the process of measurement itself alters the quantum state, collapsing it into one of the possible eigenstates.

Thus, from a single measurement, you cannot distinguish whether:

• The particles were already anti-correlated from the moment of their creation (e.g., due to conservation laws),
  or
• The SGA forced the particles into anti-correlated spin states during measurement.

3. Statistical Analysis of Many Pairs

When you measure many entangled electron-positron pairs, patterns emerge that reveal the underlying quantum correlations.

Key Observations:

• The entangled electron-positron pairs are created in a quantum state that conserves angular momentum. For example, if the original photon had spin $+1\hslash$, the spin of the electron-positron system must sum to $+1\hslash$, including any orbital angular momentum contributions.
• If many pairs are measured, the outcomes for the spins of the electron and positron will show perfect anti-correlation along the same measurement axis:
  • Whenever the electron is measured spin-up ($+\hslash /2$), the positron will be measured spin-down ($-\hslash /2$).
  • Whenever the electron is measured spin-down ($-\hslash /2$), the positron will be measured spin-up ($+\hslash /2$).

The Statistical Result:

• If you collect a large number of measurements, you will find that the up and down spin states are equally distributed across the electrons and positrons.
• However, the anti-correlation between the electron’s spin and the positron’s spin will always hold (within experimental uncertainty). This is a direct consequence of their entanglement.

4. What This Means for Anti-Correlation

1. The anti-correlation between the spins of the electron and positron is not “created” by the SGA. Instead, it reflects the entangled nature of the quantum state in which the particles were created.
2. The SGA merely reveals the anti-correlation by forcing each particle into a definite spin state along the chosen measurement axis.
3. The observed equal distribution of up and down spins in the measurements confirms that the SGA is not introducing a bias—it is simply measuring the spin components along the chosen axis.

5. Why This Matters for Entangled Pair Production

In entangled pair production, the electron and positron are created in a state that conserves angular momentum. For example:

• If the original photon had spin $+1\hslash$, the combined spin of the electron and positron (plus any orbital angular momentum) must sum to $+1\hslash$.
• This conservation law ensures that the spins of the electron and positron are correlated in a specific way, and measurement confirms this correlation.

By measuring many pairs, the anti-correlation observed in their spins provides evidence of their entangled state and confirms that the quantum state prior to measurement was consistent with conservation laws.

6. Summary

Here’s the key takeaway:

• Single Pair: For a single electron-positron pair, you cannot definitively determine whether the anti-correlation was pre-existing or imposed by the SGA during measurement.
• Many Pairs: By measuring a large number of pairs, you observe that the spins of the electron and positron are always anti-correlated along the same axis. This pattern confirms that the anti-correlation is a fundamental property of their entangled quantum state, not something introduced by the SGA.

Thus, the SGA “forces” each particle into a definite spin state, but the anti-correlation between entangled particles reflects their quantum mechanical relationship and is not an artifact of the measurement process.

1. What Does “Superimposed” Mean in Quantum Mechanics?

In quantum mechanics, a superposition means that a system exists in a linear combination of multiple possible states simultaneously, until a measurement collapses the wavefunction into one of the allowed outcomes.

For example, if an electron-positron pair is in the singlet state:

this means:

• The system is in a quantum superposition of the electron being spin-up and the positron being spin-down, and the electron being spin-down and the positron being spin-up.
• These two possibilities are not separate or independent; they are coherently combined into one quantum state.

Key point: The superposition is not just a lack of knowledge about the system’s state—it is an actual property of the system, where both possibilities coexist until measurement.

2. What Does “Unknown/Randomly Oriented” Mean?

In classical physics, saying the state of a system is “unknown” or “randomly oriented” means:

• There is a definite (but hidden) state that exists before measurement, we just don’t know what it is.
• For example, the electron might already have a definite spin-up ($+\hslash /2$) and the positron spin-down ($-\hslash /2$), but we are ignorant of this fact.
• If we measure many such pairs, the outcomes would still show anti-correlation (e.g., always opposite spins), but this would reflect a pre-existing hidden state, not a quantum superposition.

In this case, the anti-correlation arises from the fact that the particles were created with definite but unknown opposite spins, and measuring one reveals the other.

Key difference: In this classical interpretation, the randomness reflects our ignorance, not an intrinsic property of the system.

3. Why Is Superposition Different from “Unknown/Random”?

The difference lies in the predictions of quantum mechanics versus classical physics, particularly when measurements are made along different axes. This is where Bell’s theorem and the violation of Bell’s inequalities come into play.

Classical Randomness (Hidden Variables):

• If the electron and positron had pre-existing, definite spin states (e.g., electron spin-up, positron spin-down), then their measurement outcomes along any axis would follow the rules of classical probability.
• For example, if we measure the spins along axes that are not aligned (e.g., one SGA along $z$ and the other along $x$), the outcomes would still follow the classical idea of independent probabilities.

Quantum Superposition (Entanglement):

• In quantum mechanics, the entangled state predicts correlations that cannot be explained by any classical random process.
• Specifically, the measurement outcomes for entangled particles along different axes violate Bell’s inequalities, which are mathematical constraints that any classical theory (with hidden variables) must satisfy.
• This violation confirms that the particles’ states were not pre-determined or pre-existing, but instead were in a genuine quantum superposition prior to measurement.

4. Experimental Evidence for Superposition

The perfect anti-correlation of spins when measured along the same axis is consistent with both quantum mechanics and classical hidden-variable theories. However, the key difference emerges when measurements are made along different axes.

Example: Measuring Electron Spin Along $z$ and Positron Spin Along $x$:

• In classical hidden-variable theories, the outcomes would depend only on the pre-determined spin states. The correlations between the measurements would be weaker and would obey Bell’s inequalities.
• In quantum mechanics, the entangled state predicts stronger correlations, even when the measurements are made along different axes. These correlations violate Bell’s inequalities.

Experimental Results:

• Experiments (e.g., by Alain Aspect and others) have consistently shown that the correlations between entangled particles violate Bell’s inequalities.
• This violation rules out classical hidden-variable models and confirms that the particles were in a quantum superposition prior to measurement.

5. Why Can’t We Just Say “Unknown/Random”?

• If the spin states were merely “unknown” or “randomly oriented,” their behavior would follow classical probability rules. This would agree with Bell’s inequalities.
• However, the experimental violation of Bell’s inequalities shows that the correlations between entangled particles cannot be explained by classical randomness or ignorance.
• Instead, the results are consistent with quantum mechanics, where the particles exist in a superposition of states, and their outcomes are fundamentally correlated in a way that transcends classical intuition.

6. Summary: Superimposed vs. Unknown

7. Final Answer

The perfect anti-correlation of spins along the same axis is consistent with both quantum superposition and classical randomness. However:

• The violation of Bell’s inequalities in experiments confirms that the particles were in a quantum superposition of states, not merely in pre-existing, unknown states.
• Superposition describes a fundamentally quantum property of the system, where the particles are entangled and their states are not individually well-defined until measured. This is a key distinction between quantum mechanics and classical theories with hidden variables.