Why is Momentum a Hermitian Operator? | Quantum Mechanics

In this video, we will investigate whether the position operator and the momentum operator really are Hermitian operators. While the position operator looks fine, the imaginary unit "i" in the definition of the momentum operator seems a bit suspicious. After a complex conjugation, this i will become a minus i, so is the momentum operator really a Hermitian operator?

Contents:
00:00 Introduction
00:25 Definition: Hermitian
00:56 Position Operator
01:21 Momentum Operator

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[quahntasy]

*Doing Quantum optics course this sem. These videos are really helpful for my revision*

[wernerheisenberg]

When I was studying for my first qm exam, I sat about an hour on this problem because the complex conjugation irritated me so much.

Another thing which can be easily proven by partial integration is that the adjoint of the derivative operator d/dx is equal to -d/dx, so (d/dx)^{dagger}= -d/dx
Then it also fits with the -i in p and we get p^{dagger}= p.

[newideas]

Please make some series of videos on Representation theory for QFT.
It will highly appreciated.

[silentbubble]

Even though x and p are hermitian, I found that xp is not. So the product of two noncommuting hermitian operators is not hermitian? Is this generally true?

[alohahola]

Hello, I have a little question about propagators, propagators are green's functions, in non-relativistic quantum mechanics, for what differential equations are exactly the green's functions?? And how are exactly related with the time evolution operator? Sorry for my bad english

[thatonegirlyraj]

Recently I read a paper on space-time uncertainty principle which is being used in string theory but when I searched google scholar for a kind of "momentum-energy" uncertainty principle I found no text on it. Therefore I wonder that, can there be an uncertainty relation like this? So I wanted to know what are your views upon this question of mine. Btw again a nice video.❤