# Why do people even teach quantum mechanics without braket notation?

1. Dec 24, 2010

### Simfish

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. Dec 24, 2010

### Pengwuino

I used Griffiths and I wonder why people use bra-ket notation :)

3. Dec 24, 2010

### The_Duck

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. Dec 24, 2010

### Simfish

^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. Dec 24, 2010

### George Jones

Staff Emeritus
Does everyone think and learn in the same ways that you do?
It is particular apparent to anyone who teaches that different people think and learn in different ways.

6. Dec 24, 2010

### Simfish

^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. Dec 24, 2010

### Kurdt

Staff Emeritus
One of the major problems students have with quantum mechanics is that it is conceptually difficult. Adding the extra abstraction of bra-ket 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. Dec 24, 2010

### tom.stoer

Bra-ket notation is rather abstract, that means an extra abstraction on top of QM. But in addition sometimes an abstract bra-ket 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. Dec 24, 2010

### Simfish

Oh okay.

Are there problems where braket notation is a lot messier than non-braket 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/SHO-3. The matrix method and "doing the integrals in x-space" method are FAR messier than the braket notation method.

10. Dec 25, 2010

### tom.stoer

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

$$\langle k^\prime | k\rangle = 2\pi\delta_{kk^\prime}$$

but for x defined on the entire real line you get

$$\langle k^\prime | k\rangle = 2\pi\delta(k-k^\prime)$$

In both cases you have

$$\hat{p}| k\rangle = k| k\rangle$$

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 bra-ket notation w/o ever writing down (and defining!) the wave function

$$\psi_k(x) = \langle x|k\rangle$$

and the integrals.

11. Dec 25, 2010

### DrDu

It is quite nasty to represent anti-linear operations like time inversion in bracket formalism.

12. Dec 25, 2010

### Fredrik

Staff Emeritus
I'm not going to read all that, but the appearence of all those integrals in the messy calculation suggests to me that the difference between it and the "bra-ket" calculation is that the messy one uses the explicit definition of the inner product on L2(ℝ3) (over and over, without ever using the result that it is an inner product), while the "bra-ket" calculation pays no attention to which inner product we're dealing with. You don't need bra-ket notation to do that.

13. Dec 27, 2010

### cbetanco

Physicists use bra-ket notation and look at state vectors, chemists only know of wave functions (projecting the state vector onto the position basis). So you don't need bra-ket notation if you are doing chemistry!

14. Dec 27, 2010

### eaglelake

Quantum mechanics is a theory of linear operators. So, it helps to take a course in linear algebra before beginning quantum mechanics. The great thing about Dirac's bra-ket notation is that it was invented for quantum mechanics, so the calculations are most convenient when using it. You can use it with matrices or differential operators, with linear vector spaces or with linear function spaces. The equations always look the same. For example, calculations for any two-state system begin in the same way. It doesn't matter whether we are investigating the double-slit experiment or spin ½ particles, the bra-ket notation, "tells us what to do". Combined with a few postulates, you learn to do all the calculations the same way. However, in a function space, you ultimately have to solve differential equations and evaluate integrals. But, the bra-ket notation provides us with a consistent approach to most problems in quantum mechanics. Obviously, I recommend it to anyone studying quantum mechanics.

15. Dec 27, 2010

### Fredrik

Staff Emeritus
It's not the bra-ket notation that saves you from having to do integrals. It's the fact that

$$\langle f,g\rangle=\int_{-\infty}^\infty f(x)^*g(x)dx$$

defines an inner product. If we just use that, we rarely have to do any integrals.

Bra-ket 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 <f||g>=< |f>,|g> > for all |g>. Write <f|g> instead of <f||g>, because the extra vertical line is annoying. Define the product of a ket and a bra by (|f><g|)|h>=<g|h>|f>. Allow expressions of the form c|f> where c is a complex number to be written as |f>c. Note that this means that we can write |f><g|h> without causing confusion.

This simplifies some things, but it doesn't avoid any integrals.

Last edited: Dec 27, 2010
16. Dec 29, 2010

### Edgardo

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.

Last edited: Dec 29, 2010
17. Dec 30, 2010

### nismaratwork

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->Bra-Ket 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.