And then express them as **bras** and **kets**:

Which may, or may not, be helpful in keeping straight which rows and columns to multiply together.

Keeping in mind that, for instance:

Is:

]]>Thanks! It does. This part is exactly what I needed to see.

]]>So, inner product is a measure of how close to vectors are to being the same.

In quantum mechanics, the vectors in question would be the current quantum state vector and the eigenvector of some putative measurement that could be made on the system. The inner product is the probability amplitude, and its norm squared is the probability of getting that measurement.

Simply put, the closer the state vector is to a given eigenvector, the greater the probability of getting that eigenvector as a measurement result.

Thank you! It’s been a good exercise for me, trying to find a way to explain it sensibly. Goes a long way to helping clarify it in my own mind.

*“My question though is what it means in terms of the physics to use a bra instead of a ket?”*

It’s just a convenient mathematical notation for representing the inner product operation. The deeper question is your second one:

*“Or if we’re converting a ket into a bra just to take the inner or outer product, what does that mean in terms of the physics?”*

Geometrically, the inner product is the projection of one vector onto another. One can think of it as sort of casting a shadow. If the vectors are orthogonal, the inner product is zero — the shadow is “straight down” as if the sun was overhead and has no length. The more the two vectors point in the same direction, the longer the shadow one will cast on the other. If they point in exactly the same direction, the shadow has maximum length.

So, inner product is a measure of how close to vectors are to being the same.

In quantum mechanics, the vectors in question would be the current quantum state vector and the eigenvector of some putative measurement that could be made on the system. The inner product is the probability amplitude, and its norm squared is the probability of getting that measurement.

Simply put, the closer the state vector is to a given eigenvector, the greater the probability of getting that eigenvector as a measurement result.

Going a bit further down the rabbit hole, in Euclidean space, “orthogonal” means vectors are at 90°, and it’s easy to visualize that the projection is zero. If the angle is less than 90°, its also easy to visualize how one casts a “shadow” on the other. What’s a bit harder is when the angle is greater than 90° — mathematically (in Euclidean space) one gets the same result, but with a minus sign.

So, imagining normalized vectors with a length of one, starting with vector ** B** being identical to vector

Therefore, in Euclidean space, the inner product is also helpful in seeing when two vectors are obtuse, orthogonal, acute, or identical.

However, recalling the Bloch sphere, antipodal vectors (i.e. with an angle of 180°) are defined as orthogonal, so in some sense angles between them can never be obtuse. Therefore the inner product only grows from 1.0 (with identical normalized vectors) to 0.0 (with orthogonal vectors).

As an example, consider the |**0**⟩ and |**+**⟩ states:

Which is the probability amplitude of those states. The norm squared is:

And, indeed, we’d expect the probability of getting the |**0**⟩ state — assuming the system is in the |**+**⟩ state — to be 1/2.

This is also why we always get the same result making a second identical measurement. Making the first measurement puts the system in a known state, say it is |**0**⟩, and then the probability amplitude of measuring the |**0**⟩ state is:

The norm squared, of course, is 1. We’re guaranteed to get the same measurement. On the other hand, the probability of getting the |**1**⟩ state is:

So it never happens. Measurements between those states will have some non-zero amplitude less than 1.0, and will have some norm squared probability of occurring.

Make sense?

]]>My question though is what it means in terms of the physics to use a bra instead of a ket? Or if we’re converting a ket into a bra just to take the inner or outer product, what does that mean in terms of the physics? (I’ve read enough about entanglement to glean why tensor products are useful.)

No worries if this is a bottomless rabbit hole you’d rather not go down.

]]>If there are any other topics (or any questions), now would be the time to suggest them (or ask them).

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