4 ms·

I was with you until you said bra and ket didn't do it for you. Its right there in the name bra(c)ket. <bra|, |ket>. Logically equiv to any other notation, but

I was with you until you said bra and ket didn't do it for you. Its right there in the name bra(c)ket. <bra|, |ket>. Logically equiv to any other notation, but impossible to fudge up. When they are multiplied in a way that lines up like <bra|ket>, you got an inner product (scalar measure of overlap). When they are multiplied wrong way around, you got a ketbra |i><j| (which is basically a neat way to represent a matrix element at indices i and j). Is the bra the row and the ket the column or vice versa? Don't matter. Just keep things facing whichever way they were originally facing and nothing will go wrong. In fact, you don't even need to think in terms of columns at all if you don't want. That's it. That's the whole system. For the engineers who never seen this, |i> |j> |k> work exactly like unit vectors i,j,k, but you aren't limited to the usual 3 dimensions and 3 letters. Writing |x> in a ket like this just means "I am using x as a label for a degree of freedom in some vector space, not as the literal value x". This lets you do cool shit, like if you got n dimensions you can write a vector as sum v_i |i> ranging from 0 to n. You can also do super useful stuff like merging two vector spaces together to represent a larger system (aka a tensor product). Eg if I have a vector v = sum v_i |i> representing one subsystem and another vector w = sum w_j |j> representing some other subsystem, I can construct the vector u = sum v_i w_j |i, j> representing the whole system (sum is over both i and j). (I know you already learned all this, I wrote it for everyone else's benefit). Unrelated, given the webs origin at cern, HTML should have used bras as start tags and kets as end tags. <HTML| <head|head> <body| ... |body> |HTML>.

azalemeth 4y ago
Fun anecdote: when I was a teenager learning LaTeX, I'd got in the habit of google image searching the symbols – "Latex infinity" would turn up a pic of an infinity symbol with "\infty" below it; "latex contour integral symbol" would show "\oint", and so on. As a nineteen year old first-year university student, I genuinely typed "latex bra" into google image search, pressed enter, and then had a second or two of shock as my brain was genuinely surprised to not see quantum mechanics in front of me with a nifty little "\langle" involved, but rather lots of women in decidedly unique looking underwear...
- sam-s 4y ago
  I learned that "garbage collection" is not just a memory management technique but also refers to big trucks removing the content of trash cans when I took a "nerd test".
  - dbcurtis 4y ago
    One of the architects of the Pentium Pro, Intel’s first out-of-order execution engine, told of walking up to a Coke machine with an “out of order” sign on it — “Oh, out-of-order has other meanings.”
    - IIAOPSW 4y ago
      The vending machine at intel drops a can of coke. A CPU architect walking by picks it up then inserts his money. The machine is out of order.
sfpotter 4y ago
If you define u to be a column vector, and u' to be its (conjugate) transpose, and define matrix multiplication as usual, then u'v is a dot product and uv' is an outer product. What you mentioned about the standard basis vectors in R^3 works for standard basis vectors e_i in R^n using this notation; e.g., e_i e_j' is an all zero matrix with a one in the ith row and jth column. If you have a hard time remembering whether u should be a column vector, you can simply identify things that look like u with vectors (|ket>), and things that look like u' with linear functionals (<bra|). Then u'u is just a dot product. There are other notations, too... Sometimes (u, v) and u \cdot v are inner products, sometimes u \otimes v is an outer product. Generally linear algebra-style notation is terser and has less line noise. IMO, some manipulations are simpler to think about using this notation than with the bra-ket notation, especially when you're doing numerical optimization or numerical linear algebra. Maybe another reason to prefer it is that modern notation for operator algebra follows this notation and not the bra-ket notation (although I think bra-ket notation was invented for this purpose...). If L is a linear operator, L^* is its adjoint, Lu is the application of L to u, etc... Of course, what notation you should use is determined by what community you belong to. If you're publishing papers on numerical linear algebra, it would be very strange to decide to use the bra-ket notation instead. The purpose of notation is communication. The goal is to communicate ideas to other researchers. It's good to develop the ability to be flexible with notation and go along with what other people are doing.
captainmuon 4y ago
I am actually a fan of bra-ket notation. But in my mind, <i|j> is the amplitude for going from state i -> j. My point is, if you just have one state that you want to talk about, which one do you write down <a| or |a>? It doesn't matter. You only write the brakets when you want to calculate transition amplitudes/inner products, or cross products. I agree about just keeping them the way they are originally facing and nothing will go wrong. The nice thing is that you can mechanically rearrange things almost like regular algebra, i.e. draw out a common factor, and don't have to think what these objects are (a vector? a dual vector? a functional?). Same with four-vector notation / Einstein summation. As long as you are following the rules for raising and lowering indices, you cannot form "wrong" products or mess up the transformation properties. Said in another way: as a (former) experimentalist, I don't care if the vector is p^mu or p_mu. All I care about is the product p^mu p_mu.

azalemeth 4y ago

Fun anecdote: when I was a teenager learning LaTeX, I'd got in the habit of google image searching the symbols – "Latex infinity" would turn up a pic of an infinity symbol with "\infty" below it; "latex contour integral symbol" would show "\oint", and so on. As a nineteen year old first-year university student, I genuinely typed "latex bra" into google image search, pressed enter, and then had a second or two of shock as my brain was genuinely surprised to not see quantum mechanics in front of me with a nifty little "\langle" involved, but rather lots of women in decidedly unique looking underwear...

sam-s 4y ago
I learned that "garbage collection" is not just a memory management technique but also refers to big trucks removing the content of trash cans when I took a "nerd test".
- dbcurtis 4y ago
  One of the architects of the Pentium Pro, Intel’s first out-of-order execution engine, told of walking up to a Coke machine with an “out of order” sign on it — “Oh, out-of-order has other meanings.”
  - IIAOPSW 4y ago
    The vending machine at intel drops a can of coke. A CPU architect walking by picks it up then inserts his money. The machine is out of order.

sfpotter 4y ago

If you define u to be a column vector, and u' to be its (conjugate) transpose, and define matrix multiplication as usual, then u'v is a dot product and uv' is an outer product. What you mentioned about the standard basis vectors in R^3 works for standard basis vectors e_i in R^n using this notation; e.g., e_i e_j' is an all zero matrix with a one in the ith row and jth column. If you have a hard time remembering whether u should be a column vector, you can simply identify things that look like u with vectors (|ket>), and things that look like u' with linear functionals (<bra|). Then u'u is just a dot product. There are other notations, too... Sometimes (u, v) and u \cdot v are inner products, sometimes u \otimes v is an outer product. Generally linear algebra-style notation is terser and has less line noise. IMO, some manipulations are simpler to think about using this notation than with the bra-ket notation, especially when you're doing numerical optimization or numerical linear algebra. Maybe another reason to prefer it is that modern notation for operator algebra follows this notation and not the bra-ket notation (although I think bra-ket notation was invented for this purpose...). If L is a linear operator, L^* is its adjoint, Lu is the application of L to u, etc... Of course, what notation you should use is determined by what community you belong to. If you're publishing papers on numerical linear algebra, it would be very strange to decide to use the bra-ket notation instead. The purpose of notation is communication. The goal is to communicate ideas to other researchers. It's good to develop the ability to be flexible with notation and go along with what other people are doing.

captainmuon 4y ago

I am actually a fan of bra-ket notation. But in my mind, <i|j> is the amplitude for going from state i -> j. My point is, if you just have one state that you want to talk about, which one do you write down <a| or |a>? It doesn't matter. You only write the brakets when you want to calculate transition amplitudes/inner products, or cross products. I agree about just keeping them the way they are originally facing and nothing will go wrong. The nice thing is that you can mechanically rearrange things almost like regular algebra, i.e. draw out a common factor, and don't have to think what these objects are (a vector? a dual vector? a functional?). Same with four-vector notation / Einstein summation. As long as you are following the rules for raising and lowering indices, you cannot form "wrong" products or mess up the transformation properties. Said in another way: as a (former) experimentalist, I don't care if the vector is p^mu or p_mu. All I care about is the product p^mu p_mu.