Significance of the Hat Symbol in Quantum Mechanics and Classical Kinematics

[amorrow]

This is a beginner's question: what does that hat about the H (Hamiltonian) symbol signify? One often sees it in the Schrödinger equation and a few other symbols such as the p of momentum.

 

[jcsd]

This is a beginner's question: what does that hat about the H (Hamiltonian) symbol signify? One often sees it in the Schrödinger equation and a few other symbols such as the p of momentum.

Two different things is the answer. The caret above the Hamiltonian tells you that it's an operator, the caret above a momentum vector tells you that this specific vector is a unit vector.

 

[Fredrik]

Are you sure about that last one? $\hat p=-i\hbar\frac{d}{dx}$ is certainly an operator. The hat is there because the author wants to be able to write things like

$$\hat p e^{ipx/\hbar}=pe^{ipx/\hbar}$$

Amorrow, a lot of books don't use the "hat" notation. For example, Weinberg writes this $\hat p$ as P instead, and Sakurai writes it as p. Sakurai writes p' for what I called p above.

 

[jcsd]

Yes, if the the momentum is the momentum operator, then the caret means it's an operator. Tbh actually thinking about it, it's far more likely that the symbol amorrow saw was meant to represent the momentum operator.

 

[amorrow]

Yes, if the the momentum is the momentum operator, then the caret means it's an operator. Tbh actually thinking about it, it's far more likely that the symbol amorrow saw was meant to represent the momentum operator.

Yes, I was looking at the Wikipedia entry on momentum operator, etc. Their use of the caret/hat is somewhat inconsistent.

 

[jcsd]

Yes, I was looking at the Wikipedia entry on momentum operator, etc. Their use of the caret/hat is somewhat inconsistent.

Yep, in a quantum mechanical setting it's going to mean the momentum operator. In a classical kinematic setting it'd probably mean a momentum unit vector.