Is the Order of Bras and Kets Important in Equations?

[dyn]

When re-arranging equations with bras and kets I was under the impression that the order of the bras and kets had to be maintained unless they formed an inner product ie. just a complex number in which case they could be moved around ? Is this the case ? As I am confused about the following equation I found
Tr |u><u|v><v| = <v|u><u|v>
In this example the bra <v| seems to have jumped from the end to the start.

 

[Fredrik]

$$\operatorname{Tr}|u\rangle\langle u|v\rangle\langle v| =\sum_i \langle i|u\rangle\langle u|v\rangle\langle v|i\rangle = \sum_i \langle u|v\rangle\langle v|i\rangle\langle i|u\rangle = \langle u|v\rangle\langle v|\bigg(\sum_i |i\rangle\langle i\bigg)|u\rangle =\langle u|v\rangle\langle v|u\rangle$$

 

[dyn]

Thanks for that but I'm confused about the first step. At first I thought I was the imaginary number but it looks like an integer you are summing over but I don't understand how it can be brought into the equation.

 

[Fredrik]

Thanks for that but I'm confused about the first step. At first I thought I was the imaginary number but it looks like an integer you are summing over but I don't understand how it can be brought into the equation.

Sorry about that. Yes, it's an integer. I should at least have started with this statement: Let $\{|i\rangle\}_{i=1}^\infty$ be any orthonormal basis.

If the Hilbert space is finite-dimensional, replace $\infty$ with $\dim\mathcal H$ (where $\mathcal H$ denotes the Hilbert space). The first step in that calculation is just the definition of the trace of a linear operator.

You may want to also take a look at https://www.physicsforums.com/showthread.php?t=694922 about the relationship between linear operators and matrices.

 

[dyn]

Thanks. I understand it now. Could you just confirm for me that my original statement is correct , ie the order of bras and kets does matter and they can only be rearranged in order if they form an inner product ?

 

[Fredrik]

Yes, it's correct. If you move them around, you may end up with an entirely different object than the one you started with, as in this case, where |u><u|v><v| is a linear operator and <u|v><v|u> is a number.