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Why do people even teach quantum mechanics without braket notation? 
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#1
Dec2410, 01:55 AM

PF Gold
P: 828

To me, braket notation just seems much easier and more intuitive than the approach from Griffiths. And yes, I learned QM through a text that used braket notation.



#2
Dec2410, 02:25 AM

PF Gold
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#3
Dec2410, 03:16 AM

P: 865

For Griffiths you basically only need calculus. I think to get comfortable with bras and kets you need to spend some time on linear algebra.



#4
Dec2410, 03:25 AM

PF Gold
P: 828

Why do people even teach quantum mechanics without braket notation?
^That's a good point, but I'd expect that the vast majority of undergrads would have had linear algebra by the time they take quantum



#5
Dec2410, 03:42 AM

Mentor
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#6
Dec2410, 04:00 AM

PF Gold
P: 828

^Okay true, but I'd at least like to know why people prefer QM without braket notation? Are there working physicists who do?
There certainly is a reason why all the grad lvl quantum texts use braket notation. 


#7
Dec2410, 06:15 AM

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PF Gold
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One of the major problems students have with quantum mechanics is that it is conceptually difficult. Adding the extra abstraction of braket notation at the beginning may hinder the learning process. It therefore seems logical to me to stick with a notation that is very familiar.
Besides it depends on the aims of the course. Some are taught from the historical perspective rather than the approach laid out in Sakurai's text for example, with the benefit of hindsight. I guess what I'm saying in summary is that it eases students into QM before bombarding them with more abstarction than they need. 


#8
Dec2410, 07:58 AM

Sci Advisor
P: 5,451

Braket notation is rather abstract, that means an extra abstraction on top of QM. But in addition sometimes an abstract braket Hilbert space is not sufficient; instead one has to deal with functions (depending on x, k, ...) in order to study the probability density, scalar product, convergence, asymptotic behaviour, ...
So essentially we need both. 


#9
Dec2410, 12:46 PM

PF Gold
P: 828

Oh okay.
Are there problems where braket notation is a lot messier than nonbraket notation? It seems that braket notation makes some problems A LOT cleaner, which in turn, makes them easier for me to understand. Here's an example (with a simple harmonic oscillator): http://www.scribd.com/doc/45874729/SHO3. The matrix method and "doing the integrals in xspace" method are FAR messier than the braket notation method. 


#10
Dec2510, 04:34 AM

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P: 5,451

If you try to solve am simple problem like eigenstates k> of the momentum operator p you immediately face the problem of normalization and inner product. If x is a compact variable [0,L] you find
[tex]\langle k^\prime  k\rangle = 2\pi\delta_{kk^\prime}[/tex] but for x defined on the entire real line you get [tex]\langle k^\prime  k\rangle = 2\pi\delta(kk^\prime)[/tex] In both cases you have [tex]\hat{p} k\rangle = k k\rangle[/tex] but the space of kets is different. In the first case it's related to the L²[0,L] Hilbert space of square integrable functions, but in the second case you have to discuss generalized functions (distributions), Sobolev spaces and all that. You are not able to do this based on an abstract braket notation w/o ever writing down (and defining!) the wave function [tex]\psi_k(x) = \langle xk\rangle[/tex] and the integrals. 


#11
Dec2510, 04:47 AM

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It is quite nasty to represent antilinear operations like time inversion in bracket formalism.



#12
Dec2510, 07:12 AM

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PF Gold
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#13
Dec2710, 02:01 PM

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#14
Dec2710, 08:00 PM

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#15
Dec2710, 08:45 PM

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PF Gold
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It's not the braket notation that saves you from having to do integrals. It's the fact that
[tex]\langle f,g\rangle=\int_{\infty}^\infty f(x)^*g(x)dx[/tex] defines an inner product. If we just use that, we rarely have to do any integrals. Braket notation is defined by the following: Write f> instead of f. This turns <f,g> into < f>,g> >. (This is still the inner product of two vectors, now written as kets). Let <f be the linear functional defined by <fg>=< f>,g> > for all g>. Write <fg> instead of <fg>, because the extra vertical line is annoying. Define the product of a ket and a bra by (f><g)h>=<gh>f>. Allow expressions of the form cf> where c is a complex number to be written as f>c. Note that this means that we can write f><gh> without causing confusion. This simplifies some things, but it doesn't avoid any integrals. 


#16
Dec2910, 08:25 PM

P: 685

The article Mathematical surprises and Dirac's formalism in quantum mechanics by François Gieres explains why Dirac's notation can cause mathematical problems (links: arxiv and iopscience).
See chapter 4.2. Chapter 5 mentions literature that differ in the mathematical rigor. 


#17
Dec3010, 06:23 AM

P: 2,284

I think that it's worth remembering who Dirac was... not a man who made this system for anyone's use but his own. That it became widely adopted is more a function of the times, and what he contributed to QM. It's hardly perfect, as others have noted, nor can any single formal notation fill all needs. As someone trying to take the Linear Algebra>BraKet move, I find it to be just... much more of the same. Plenty of advantages, and some notable problems.
I think a better question might be why anyone should be wedded to one notation when it's very specific in its application to QM. Or, to put it another way... would you learn shorthand first, or the language that shorthand is based on? Generally most theories of how people learn are based on building upon previous ideas... you would offer an abstraction of an abstraction of an abstraction... without the background? Seems like the ultimate preparation for "shut up and calculate" to me. 


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