[Physics] Advantages of using bra-ket notation

hilbert-spacenotationquantum mechanics

I’m curious whether people use bra-ket notation in QM for any reasons beyond convention.

Are there any advantages to using bra-ket notation over ordinary linear algebraic notation? Are certain operations relevant to QM represented more compactly in bra-ket notation? Or does bra-ket notation clarify relationships between certain linear algebraic concepts?

Best Answer

Bra-ket notation is useful because it lets you get rid of excess superscripts and subscripts.

For example, in conventional vector notation you might call the unit vectors in 3D space $\vec{e}_x$, $\vec{e}_y$, and $\vec{e}_z$. There's something like "$\vec{e}$" that you "hang an index on", in order to specify which basis vector you mean. But in bra-ket notation, you can just write the index by itself, as $|x \rangle$, $|y \rangle$, and $|z \rangle$. This is a small space savings in this case, but it gets very useful when there are multiple indices. For example, the state of three independent particles could be written as $$|\mathbf{x}, \mathbf{y}, \mathbf{z} \rangle$$ in bra-ket notation, while without it you would need something annoying like $$\vec{e}_{\vec{x}} \otimes \vec{e}_{\vec{y}} \otimes \vec{e}_{\vec{z}}.$$ You can't ever get rid of the $\vec{e}$ in that notation, because then it would be ambiguous whether $\vec{x}$ by itself is a position vector or a state. It gets even better when you start considering identical particles. If they're bosons, then in completely rigorous mathematical notation you would have to write something like $$\frac{1}{\sqrt{6}} \left( \vec{e}_{\vec{x}} \otimes \vec{e}_{\vec{y}} \otimes \vec{e}_{\vec{z}} + \vec{e}_{\vec{x}} \otimes \vec{e}_{\vec{z}} \otimes \vec{e}_{\vec{y}} + \vec{e}_{\vec{y}} \otimes \vec{e}_{\vec{z}} \otimes \vec{e}_{\vec{x}} + \vec{e}_{\vec{y}} \otimes \vec{e}_{\vec{x}} \otimes \vec{e}_{\vec{z}} + \vec{e}_{\vec{z}} \otimes \vec{e}_{\vec{x}} \otimes \vec{e}_{\vec{y}} + \vec{e}_{\vec{z}} \otimes \vec{e}_{\vec{y}} \otimes \vec{e}_{\vec{x}} \right)$$ while in physics we just continue to write $|\mathbf{x}, \mathbf{y}, \mathbf{z} \rangle$. The reason we can do that is because it is understood that in Dirac notation, the only purpose of the stuff inside the $|$ and $\rangle$ is to act as a descriptor that makes it clear which state we're talking about in the present context. This allows us to suppress irrelevant details, which is essential for doing real calculations.

Another benefit is when you start using bras. For example, you can form things like $$\langle \mathbf{x} | \mathbf{y} \rangle, \quad | \mathbf{x} \rangle \langle \mathbf{y}|.$$ From the shape alone you can easily tell that the first is a number and the second is a linear operator. Of course, you could do the same thing in vector notation with conjugate transposes, $$\vec{e}_{\vec{x}}^\dagger \vec{e}_{\vec{y}}, \quad \vec{e}_{\vec{x}} \vec{e}_{\vec{y}}^\dagger$$ but this will probably give you wrist strain writing it and eye pain reading it. (Sometimes people just write $\vec{e}_{\vec{x}} \vec{e}_{\vec{y}}$ for the latter and call it a "dyadic", but I think this is even worse because it's ambiguous inside larger expressions.) Dirac notation looks even better when you consider conjugation. In Dirac notation this is done for bras and kets by flipping everything horizontally, while for vector notation you have to add and remove daggers.

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