What are Product States?

Product States are states of entangled particles – or particles that have interacted in some manner (some energy exchange). They are ‘correlated’ in this sense.

They are written as the product of state 1 and state 2 – but given that this is quantum mechanics, we have to consider the superposition of all such products.

So – | 1> |2> + |2> |1>   – would be the superposition of states 1 and 2.

How are these different from classically correlated states?

If you know the combined (correlated state) in classical physics, you also know  everything about the INDIVIDUAL component states.

Here is an example (borrowed from Susskind’s Quantum Mechanics):

If Joe hands out two dice – one to Alice and one to Bob – these dice are correlated – in that they came from the same source. Before handing them out, Joe peeks at the entire state – i.e. he sees what is showing on DICE 1 and what is showing on DICE 2.

Now, no matter how far apart Alice and Bob take their individual die, Joe will always know what  the individual reading on each die was.  So – knowledge of the ORIGINAL COMBINED state – also leads to DIRECT KNOWLEDGE of the individual states.

This is JUST NOT TRUE in Quantum Mechanics.

Why is it difficult to visualize these product states?

Even the simplest SINGLE particle state  (in QM) is a 2 state system. So – a single particle is itself a multi-state system. Now, think about more than one particle – and the number of particle states. Instead of individual particle states, there is now a ‘combined state’ of the multiple particles.

What does the product state operator act on? Individual states or a product state?

Are measurements the same as Operating on a state?

In general, No. Only when the state is actually in an eigenstate of the operator, can we say that ‘measuring’ the state is the same as operating on a state vector.

Prepared states vs. Unknown states

This is at the heart of a lot of QM confusion when speaking about Operators and States. States  can be KNOWN in advance of a measurement (i.e. we PREPARE the state to be a specific state). This can be PREPARED For either single particle or MULTI particle states.

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Trying to simulate a Quantum System on a classical Turing Machine is not possible (see Feynman’s Papers below).

Enter David Deustch

Deutsch proposed the first design for a computer that operated quantum mechanically. The building block circuits were all designed as quantum circuits.

David Papers on Quantum Computing

The Universal Quantum Computer

Feynman’s Papers on Computation

1. 1960: There’s Plenty of Room at the Bottom.
   https://resolver.caltech.edu/CaltechES:23.5.1960Bottom
2. 1982: Simulating physics with computers.
   https://doi.org/10.1007/BF02650179
3. 1985: Quantum Mechanical Computers.
   https://www.optica-opn.org/home/articles/on/volume_11/issue_2/features/quantum_mechanical_computers/

Turing Machines

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As per Penrose

The current big bang model, which is partially grounded in inflation, doesn’t supply a reason as to why a low entropy, highly ordered state existed at the birth of our universe. That is, UNLESS things were set in motion long before the big bang actually occurred. In Penrose’s theory, our universe has, and will again, return to a state of low entropy as it approaches its final days of expanding into eventual nothingness, leaving behind a cold, dark, featureless, abyss.

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• Frequency is related to the energy of the photon. Higher frequency photons have higher energy and vice versa.
• Low energy photons (such as radio photons) behave more like waves, while higher energy photons (such as X-rays) behave more like particles.

What happens in High Energy Collisions? Are photons powerful enough to form electron positron pairs?

In high energy collisions, photons are emitted. Sometimes, the energy of the photons is large enough that it can b e

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De Broglie actually proposed that ALL matter was essentially waves (of what material – he did not conjecture).  The waves all ‘grouped’ together close to a point that we consider to be the material particle.

This is different from Schrodinger’s wave for a particle – which is neither real – nor does it represent the actual particle’s position.

However, both these physicist were correct. De Broglie’s waves can be applied to photons and other massless particles – as their position essentially coincides with the grouping of their ‘waves’

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For electrons and other material particles, this begs the question – what is E in the equation below?

1. Is E equal to the rest energy ?
2. Or is it equal  to the rest energy plus the binding energy (electron in an atom)?
3. Or it is equal to rest energy plus binding energy plus it’s kinetic energy?

Actually, it could be any of the three above. Hence, one must be careful what one means by E=h\nu when applied to electrons and other mass bearing particles.

The Resolution?

The actual E in the equation above is not important. What’s important are the Energy differences. When the electron jumps from an allowed state to another allowed state, it emits photons of energy equal to E. And that is exactly what the E in the   represents for an electron.

The Fundamental disagreement between Relativity and QM

I spent a lot of time studying this paper on the Fundamental Disagreement of Wave Mechanics with Relativity

These kind of paradoxes can be resolved when one keeps in mind that the E above is not really the energy of the material particle, but rather, the energy emitted when the particle changes between states. The paper above is still legit – and highlights a slightly different issue.

Pilot wave Theory DeBroglie

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To probe smaller and smaller particles, requires smaller and smaller wavelengths. Which means, higher and higher momenta. Which means higher and higher energy.

This is why , to probe the smallest of particles, we need our high energy accelerators.

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It was obviously going to be the wave function amplitude. Amplitudes take the place of probabilities, in Quantum Mechanics. This single mathematical fact is responsible for ALL the weirdness that comes about (interference, entanglement…).

Feynman proceeded – if amplitudes are to be added, how about adding ALL the amplitudes for a particle to get from A (emitter) to B (screen)? This seemed like a ridiculously infinite sum, but Feynman proceeded undaunted. And sure enough, several amplitudes canceled themselves out  – leading to the now famous Feynman Integral approach to Quantum Mechanics.

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