Formalism: Observables and Hermitian Operators

In Chapter 1, I think I said that the big beef between Einstein’s side and the Copenhagen side was the concept of ‘observation’,

the classical ‘physical quantity’ called ‘Q’ is now

in quantum mechanics an operator

and what this means is ’the measurement of the physical quantity’!!!

Which is to say the result yields its ’expectation value’.

Writing that ‘observation’ as a formula,

expectation of Q =

earlier on I was going on about how this is like, the physical-quantity-measuring guy. Like, an energy measurement request;;;;

I was spouting this kind of bullshit… haha.

The above is the definition, the definition, the definition haha.

So to do a Measurement, mathematically amounts to ‘multiplying by an n x n matrix’.

But the expectation value that comes out that way — (common-sensically) would be a real number, right!?!?

Now I’m going to prove this,

‘Whaaat~ how is it that so obviously, every single~ one of them comes out as a real number?

Once in a while a complex number could show up too~~’

Writing ’that value, as a result, is always real’ as a formula,

it comes out like this.

Now I’m going to write this equation out going down,

In conclusion,

the operator Q that was on the right ket, being on the left bra — the computation result in this case has to be the same~~~ that’s what it’s saying,

uh-huh-huh-huh-huh, and for an operator that satisfies that red thing, the math folks have a separate name

“Hermitian Operator” they call it! hahaha!

Hermitian means

something satisfying this is called a Hermitian (matrix)!!!.

this is called a dagger

(so the shape it has — actually, it really is meant to depict a dagger, a knife lolololol)

Taking the dagger, in matrix terms, means

applying Transpose & Complex conjugate!!

But in Dirac notation, the relationship between a bra vector and a ket vector

is taking the transpose and the complex conjugate, right?!?!?! isn’t it?!!? (You can tell if you look at what I wrote earlier.)

Conclusion:

“Aha!!! Measuring means multiplying by a matrix,

and we can say it’s a meaningful result only if it comes out as a Real real number.

Whaaat, but the operator that does the measuring is not just any arbitrary matrix — it turns out to be a Hermitian matrix~~~~~~!!~~!~!!!!”

“Aha, in quantum mechanics operators are aaa~~~~~lll~~~~~ Hermitian Operators.”