Mastering Bra-ket Notation: Frustrations with Quantum Mechanics Lectures

[robertjford80]

I'm a little frustrated with the quantum m lectures I've been watching. I've watched Susskind's, one in India and now James Binney's, as well as read about 3 books. They all teach this Bra-ket notation and in none of the three books I have on worked problems do they every give you a chance to practice these principles. Is it because Bra-ket notation is so easy? Am I supposed to have already mastered it in some other mathematics course? I've gotten up to Binney's fourth lecture and I can't even do one QM problem yet. This is crazy. All of the principles he's outlined on the chalkboard, nowhere have I been given any opportunity to practice them. I also looked on the web putting in google bra-ket notation problems and answers and i couldn't find anything.

I'd like to know why am I not being tested on the principles introduced? It seems basically axiomatic that when you introduce a principle in phys or math that you then test the reader to check if they understand.

 

[jedishrfu]

you can get some background here, its basically an abbreviation of an inner product so that <A|B> is A* times B where A* is the complex conjugate of A.

http://en.wikipedia.org/wiki/Bra-ket_notation

and also this ebook may help:

http://farside.ph.utexas.edu/teaching/qm/lectures/lectures.html

 

[robertjford80]

I don't need more explanation of what Bra-ket notation is. I have about 10 books on QM. What I need to do is practice. I can't learn math principles unless I practice them.

 

[marcusl]

You have ten books on QM and you can't find any problems using bra-ket notation? Have you looked at the problems after each chapter?

You say you can't do any problems in QM, so I suspect your problem is not with bra-ket formalism but with the concepts of QM. Most texts teach Schrodinger's wave mechanics first before Heisenberg's somewhat more abstract matrix formalism (of which bra-ket is just mathematical notation as jedishrfu said) so you need to get that down first. Also, 10 texts and 3 lecture series are way too much, you can't learn from such a muddle of different materials. Pick one lecture, and use the book and the recommended supplementary text specified in the syllabus. If these lectures aren't structured that way, then get one that is (like MIT's OCW series). Put the rest of your books in the back of the closet. Study wave mechanics until you understand it before moving on.

Third, you can't do QM without a solid math background. Sorry, but there's no shortcut. Make sure you understand and can work problems in complex variables and Fourier transforms, and have *mastered* the following courses: ODE's, PDE's, and linear algebra. Then you are ready to study QM.

 

[robertjford80]

You have ten books on QM and you can't find any problems using bra-ket notation? Have you looked at the problems after each chapter?

Sure there are problems but no one is showing me how to do them. It's like saying, here, translate انا مش بدي بروح الا فلسطين without give me a dictionary or a grammar.

Can you translate that? No, you can't translate that because you've never been shown how to do it step by step. I have the same problem with bra-ket notation. The problems in the back of the chapter are useless unless someone shows me how to do them. As of yet however I have seen no problems that specifically relate to braket notation. James Binney's lectures at the moment seem the best since he has a book to accompany them but the problems he has for some reason he doesn't have the solutions until he gets to chapter 3 sadly.

Third, you can't do QM without a solid math background. Sorry, but there's no shortcut. Make sure you understand and can work problems in complex variables and Fourier transforms, and have *mastered* the following courses: ODE's, PDE's, and linear algebra. Then you are ready to study QM.

I have a pretty good understanding of these math concepts but admittedly shaky. I was going to repractice the math as I encountered them in QM, that way I don't waste time practicing something that I don't need.

 

[HJ Farnsworth]

Sure there are problems but no one is showing me how to do them.

You should get a solutions manual to one of your texts in that case. Considering that you have other texts, you could use one for examples and the other for practice problems or something.

-HJ Farnsworth

 

[Fredrik]

You don't need to master two courses on differential equations before you begin. You need to understand what a differential equation is, and it helps to have some experience solving differential equations, but even that isn't absolutely essential. Linear algebra is more important. I like the book by Axler.

If you understand bra-ket notation (e.g. what I said in this post), then it is very easy, since all the "products" are defined so that you can pretend that you're dealing with a single associative multiplication operation. Sakurai ("Modern quantum mechanics") uses bra-ket notation through the entire book, and contains lots of problems, so maybe you could try to solve a few problems there.

 

[stevendaryl]

Sure there are problems but no one is showing me how to do them.

Well, is there a specific problem you'd like help understanding in how to do? You could try posting to the Advanced Physics Forum: https://www.physicsforums.com/forumdisplay.php?f=154

 

[robertjford80]

You should get a solutions manual to one of your texts in that case. Considering that you have other texts, you could use one for examples and the other for practice problems or something.

-HJ Farnsworth

I've got three books which are devoted entirely to problems and solutions (there's probably 1500 worked problems in total since one of them has 1200 problems, it's 800 pages long) but none of them deal with the nitty gritty of bra-ket notation. I'm starting to believe that it's just assumed that bra-ket notation is so easy that it's not worth working on. All of the three books I have (of worked problems) start with wave equation problems but there is a lot that is assumed that I cannot figure out how to master.

If I can't get anywhere in two weeks I'm going to have to break down and pay 300 dollars for hopefully 10 hours of private tutoring, that's how desperate I am to learn QM.

 

[HJ Farnsworth]

Hmm... well you're probably having one of two problems - either your background isn't quite as strong as it needs to be (which I think could be fixed fairly easily), or you've allowed yourself to become too intimidated by the subject to learn it (that happens to me often when learning a new notation or concept).

If the problem is background:

Marcusl said you need a solid understanding of linear algebra - I would agree that that's the key. I'm using Griffiths QM (do you have that one? It might help the forum if you posted what resources you currently own. If you don't have it, I'm sure there are similar approaches online or in other books) to learn QM, and I spent a LOT of time on Chapter 3, and gave myself a lot of time to just kind of digest it. I learned more linear algebra in that chapter than I did in my linear algebra course.

In my view, the basic idea is that the notation is a powerful way to relate between thinking about QM concepts in terms of vectors and matrices, while at the same time thinking about them in terms of functions and integrals - and I would make sure you understand it well in terms of functions and integrals before tackling the notation.

The absolutely key concept is that the functions being dealt with satisfy the axioms of a vector space, and so eigenvectors=eigenfunctions. If I were you, I'd take a lot of time to understand what Hilbert spaces are and where they come from, make sure you understand Hermitian transformations, etc. But first, make sure you understand the concepts from the integral perspective - the integrals in introductory QM are essentially equivalent to the equation <f>=$\sum$P_if_i - basically every time I learned a concept from the integral perspective, I took the time to make sure I understood how it related back to that simple equation.

If the problem is intimidation:

Take a week off. You sound pretty stressed.

You mentioned a tutor also - that works greatly sometimes, so if you try everything and nothing works, and you really do want to gain a good understanding of the subject, it's probably worth forking out a little cash ($300 is nothing compared to the cost of a lot of textbooks or a college course, not to mention the opportunity cost of the time you're putting into this). I get the impression that you're self-studying (like me), but you could also try emailing some professors or grad students at a nearby school and see if they'd be willing to help you out during their office hours a little - from personal experience, some professors might get a little annoyed, but a lot actually do thoroughly enjoy teaching and will be more than happy to help someone out who is clearly determined to learn, as you are.

Best of luck.

-HJ Farnsworth

 

[kloptok]

I would recommend looking at chapter 1 in Sakurai's "Modern Quantum Mechanics". First read through carefully at least once so that you understand it, then look at the exercises (particularly those pertaining to bra-ket notation). This should help.

 

[robertjford80]

I'm just going to try to memorize every equation I encounter, even though I don't know what the equations really mean. Hopefully one day it will make sense to me.

As for Griffiths book, yea, I have that one, but there is something I call word to equation ratio. Griffith's book has few words and a lot of equations. Generally the more equations there are the more mathematical the author is and mathematicians have an annoying habit of assuming everyone understands things exactly the way they do so they skip over steps like mad. I looked at Griffiths book and the first equation assumes that you understand the Sh eq. So right there that book starts out at a level that I'm not at yet.

 

[kloptok]

At this point I think it would be a good idea for you to give a specific example of something you don't understand, an equation, a sentence or section or otherwise. If you have trouble understanding Griffiths maybe you don't have the necessary prerequisites. From the preface (which is usually a good idea to take a look at to see if this is the right book for you) to said book:

"The reader must be familiar with the rudiments of linear algebra, complex numbers, and calculus up through partial derivatives; some acquaintance with Fourier analysis and the Dirac delta function would help. Elementary classical mechanics is essential, of course, and a little electrodynamics would be useful in places."

 

[Ken G]

Why don't we just give him a problem and see if he can do it? Let's start with the idea that |s> is the notation for some given preparation of a particle, call it the state s. There's nothing to know yet, this is just a definition and a notation for it. Now let's say that we can do a measurement on the particle, and get only one of two outcomes, call them -1/2 or +1/2. The measurement does not destroy the system, it merely puts the particle into the state corresponding to -1/2, and let's call that state |-> , or +1/2, and let's call that |+> . An important aspect of quantum mechanics is we can always write any state |s> as a linear superposition of the states corresponding to all possible outcomes of some measurement, which here are |-> and |+>, so this means:
|s> = a|-> + b|+>
for a and b some complex numbers that correctly describe that superposition.
Problem: use bra and ket notation to give expressions for a and b in terms of |s> and <-| and <+|. (Note that you are not calculating the numbers a and b, you are simply using bra and ket notation to express them. bra and ket notation doesn't do calculations, it is just a good way to denote the concepts.) Then say what is the connection between the expressions you give and the common concepts from linear algebra, involving projections onto basis vectors. Finally, say what is the physical significance of the coefficients a and b in regard to the measurement of either -1/2 or +1/2.

Once you understand these answers, you are well on your way to using bra and ket notation to simplify the basic thought processes of quantum mechanics (which is their purpose). The main new wrinkle is that this problem applies when the possible outcomes of a measurement are discrete (which is always true if you think about it), but often we wish to imagine that continuous outcomes are possible (say, for a position measurement). When we are choosing to imagine the outcomes are continuous (which is never really true, but continuous approximations to discrete things is not unusual in physics), then the coefficients in the superposition are a continuous function that can be used to calculate a probability distribution rather than discrete probabilities. The probability distribution must be integrated over some finite bin, like a bin of positions dx, before it can mean an actual probability. But this isn't really that much of a complication. Note also that when outcomes of position measurements are chosen as the "basis set" for which to express the superposition, then the coefficients of the superposition are given the special name psi(x), but it's not really much different from a and b above, it's just a continuous amplitude distribution and requires integration over dx to give finite probabilities. That can be the next problem.

 

[Fredrik]

I looked at Griffiths book and the first equation assumes that you understand the Sh eq.

All introductory QM books require you to understand complex numbers and partial derivatives. None of them requires you to understand the significance of the Schrödinger equation, or how to solve it for different potentials and boundary conditions. If you knew that already, you wouldn't need the book, because that's precisely what it will explain to you.

 

[HJ Farnsworth]

I'm just going to try to memorize every equation I encounter, even though I don't know what the equations really mean.

I strongly recommend against this approach. Unless your brain is pretty much a hard drive, this will quickly become overwhelming, and pointless if you don't understand their meaning.

I looked at Griffiths book and the first equation assumes that you understand the Sh eq. So right there that book starts out at a level that I'm not at yet.

I could be wrong, but I think the way most textbooks on QM start out is by introducing you to the Schrodinger equation - but that does not mean they expect you to understand it. They will make you understand it.

Here are some helpful hints about the Schrodinger equation to start out:

1. It is a postulate, and cannot be derived from simpler principles. QM has to start out with it. (This begs the question "where did it come from?". Honestly, for the most part, it's best to ignore this question for the time being. Ultimately, it came from a bunch of previous experiments and theorizing(photoelectric effect, De Broglie wavelength, etc.), which I think (not sure though) that Schrodinger sort of combined by saying "this data can be viewed as analogous to some optics data that people have already figured out wave equations for", so he used some equations from a completely different branch of physics and showed for the first time a form of the Schrodinger equation. But again, in QM, it is a postulate).

2. In QM, it plays the same role that Newton's F=dp/dt plays in classical mechanics. It tells you how the system evolves and let's you solve for the system given boundary conditions (in F=dp/dt, the boundary conditions will be where everything is and how fast everything is moving at one time. From this, you can derive where everything will be and how fast everything will be moving at all times using F=dp/dt. The Schrodinger equation plays a similar role).

3. It is a partial differential equation. This is the reason people are saying you should have differential equations down as a prerequisite - a statement I only partially agree with. It certainly makes the going easier if you can look at some basic differential equations that will be presented and know how to solve them, but books or online sources will take you through how the relevant ones are solved. It is important that you can follow these solutions and know why they work, or at least be able to take the solution, take the appropriate derivatives, and see that it satisfies the original differential equation.

When you look at the Schrodinger equation, you do need to think "this is a differential equation with a term that is completely unknown (namely, V). A lot of what I'm going to learn in learning QM is how to solve the equation for different V's".

So don't expect yourself to understand the Schrodinger equation the first time you see it. A book on Ancient Rome will probably start off by talking about Ancient Rome - that shouldn't make you think, "Ah great, they expect me to already know about Ancient Rome!". On the contrary, they are going to teach you about Ancient Rome.

 

[HJ Farnsworth]

One more thing I should mention - you do not need bra-ket notation to understand several solutions to the Schrodinger equation. It's easier to learn some of these solutions first. This goes back to what I was saying in one of my previous posts about understanding basic QM from the integral perspective before tackling the more abstract bra-ket notation.

-HJ Farnsworth

 

[WannabeNewton]

Landau's quantum mechanics text introduces bra - ket and has solutions to problems worked out I would say in a nice fashion. You could try that. But it isn't used throughout.

 

[robertjford80]

Why don't we just give him a problem and see if he can do it? Let's start with the idea that |s> is the notation for some given preparation of a particle, call it the state s. There's nothing to know yet, this is just a definition and a notation for it. Now let's say that we can do a measurement on the particle, and get only one of two outcomes, call them -1/2 or +1/2. The measurement does not destroy the system, it merely puts the particle into the state corresponding to -1/2, and let's call that state |-> , or +1/2, and let's call that |+> . An important aspect of quantum mechanics is we can always write any state |s> as a linear superposition of the states corresponding to all possible outcomes of some measurement,

which here are |-> and |+>, so this means:
|s> = a|-> + b|+>
for a and b some complex numbers that correctly describe that superposition.
Problem: use bra and ket notation to give expressions for a and b in terms of |s> and <-| and <+|. (Note that you are not calculating the numbers a and b, you are simply using bra and ket notation to express them. bra and ket notation doesn't do calculations, it is just a good way to denote the concepts.) Then say what is the connection between the expressions you give and the common concepts from linear algebra, involving projections onto basis vectors. Finally, say what is the physical significance of the coefficients a and b in regard to the measurement of either -1/2 or +1/2.

You see right there, that's my problem. I haven't been challenged in my books with this type of problem and quizzed on it. The three books of worked problems I have assume you can already do this. Should I have learned this in linear algebra? If not, then where? It is true that I skipped through a lot of LA but that's because I'm so impatient and I rush through things because I want to understand QM quickly. I can't help myself.

 

[robertjford80]

Landau's quantum mechanics text introduces bra - ket and has solutions to problems worked out I would say in a nice fashion. You could try that. But it isn't used throughout.

Good, they have that at the library I'll take a look at it.

I think I'm going to go back review some Linear Algebra and PDE as well as probability theory for about 3 weeks before I make another stab at QM.

 

[WannabeNewton]

You see right there, that's my problem. I haven't been challenged in my books with this type of problem and quizzed on it. The three books of worked problems I have assume you can already do this. Should I have learned this in linear algebra? If not, then where? It is true that I skipped through a lot of LA but that's because I'm so impatient and I rush through things because I want to understand QM quickly. I can't help myself.

You can't skip a lot of LA and expect to learn QM quickly. That's like skipping a lot of Riemannian geometry and expecting to learn GR quickly. Anyways, both Shankar and Sakurai develop the notation from the very very beginning, and both have good problems to get you acquainted with it (Sakurai has more problems than Shankar). I don't know if doing a book of worked problems will help you as much if you don't go through a textbook in the usual way in conjunction. The first section of Sakurai itself provides a good amount of insight that I don't think a book of worked problems will immediately get you on their own. The two should complement each other.

 

[Ken G]

Should I have learned this in linear algebra? If not, then where? It is true that I skipped through a lot of LA but that's because I'm so impatient and I rush through things because I want to understand QM quickly. I can't help myself.

Let's frame the same question again, but in linear algebra terms. Let's start with the idea that |s> is the notation for some vector. There's nothing to know yet, this is just a definition and a notation for it. Now let's say it's a two-dimensional vector, and we have two basis vectors for the space, and let's call those basis vectors |-> and |+> . An important aspect of linear algebra is we can always write any 2D vector |s> as a linear superposition of the basis vectors, which here are |-> and |+>, so this means:
|s> = a|-> + b|+>
for a and b some real numbers that correctly describe that superposition.
Problem: use an inner product to give expressions for a and b in terms of |s> and the basis vectors. (Hint: you should use orthonormality conditions that come along with inner products involving basis vectors. Note that you are not calculating the numbers a and b, you are simply using inner product notation to express them. Inner product notation doesn't do calculations, it is just a good way to denote the concepts.) And I'll bet, in the process of trying to find a good way to express the appropriate inner product, you are led to bra and ket notation, especially if you think of a bra as a row vector and a ket as a column vector in terms of a more standard x and y basis.

 

[Fredrik]

Should I have learned this in linear algebra?

Some of it yes. A QM book will typically tell you that

a) the state of the system is represented by a vector in a complex Hilbert space
b) the measuring device is represented by a self-adjoint operator,
c) eigenvalues of self-adjoint operators are real numbers
d) measurement results are eigenvalues of the operator that represents the measuring device
e) a measurement that doesn't destroy the system and has result a leaves the system in a state represented by an eigenvector of the operator, with eigenvalue a.
f) eigenvectors of the operator that represents the measuring device, are orthogonal if they correspond to different eigenvalues.

The terms vector, linear operator, self-adjoint, eigenvalue, eigenvector and orthogonal, are all explained in linear algebra. If you don't understand them, you will find QM impossibly hard. Proofs of c and f can be found both in linear algebra books and QM books. The term Hilbert space actually belongs to functional analysis, not linear algebra, but it's not essential that you understand it perfectly. If you understand the concept of "inner product space" from linear algebra, it's usually enough to know that a Hilbert space is a special kind of inner product space.

To solve the problem that Ken G posted, you must understand the above, in particular f and the concept of orthogonality.

 

[lugita15]

robertjford80, I suggest Townsend's book, A Modern Approach to Quantum Mechanics. It takes great pains to explain exactly how to do all the bra-ket manipulations.

 

[robertjford80]

Some of it yes. A QM book will typically tell you that

a) the state of the system is represented by a vector in a complex Hilbert space

The term Hilbert space actually belongs to functional analysis, not linear algebra, but it's not essential that you understand it perfectly. If you understand the concept of "inner product space" from linear algebra, it's usually enough to know that a Hilbert space is a special kind of inner product space.

Thanks for your detailed help. Yes, in my text on LA they have only one page devoted to Hilbert Space. So I guess I'm going to have to look up functional analysis to figure out hilbert space. Someone told me on this thread to understand Hilbert Space very carefully.

Anyway, my current plan is study probability for 20 hours, 40 more hours of LA and 20 more hours of PDE and maybe 10 hours reviewing ODEs, I should be able to get that done in 22 days since I study for 4.2 hours a day

 

[robertjford80]

robertjford80, I suggest Townsend's book, A Modern Approach to Quantum Mechanics. It takes great pains to explain exactly how to do all the bra-ket manipulations.

Thanks for the tip.

 

[robertjford80]

Let's frame the same question again, but in linear algebra terms. Let's start with the idea that |s> is the notation for some vector. There's nothing to know yet, this is just a definition and a notation for it. Now let's say it's a two-dimensional vector, and we have two basis vectors for the space, and let's call those basis vectors |-> and |+> . An important aspect of linear algebra is we can always write any 2D vector |s> as a linear superposition of the basis vectors, which here are |-> and |+>, so this means:
|s> = a|-> + b|+>
for a and b some real numbers that correctly describe that superposition.
Problem: use an inner product to give expressions for a and b in terms of |s> and the basis vectors. (Hint: you should use orthonormality conditions that come along with inner products involving basis vectors. Note that you are not calculating the numbers a and b, you are simply using inner product notation to express them. Inner product notation doesn't do calculations, it is just a good way to denote the concepts.) And I'll bet, in the process of trying to find a good way to express the appropriate inner product, you are led to bra and ket notation, especially if you think of a bra as a row vector and a ket as a column vector in terms of a more standard x and y basis.

Thank your for putting so much effort into helping me out but I've already decided to go back and learn more of LA. I hope to be back struggling with QM soon.

 

[Fredrik]

Thanks for your detailed help. Yes, in my text on LA they have only one page devoted to Hilbert Space. So I guess I'm going to have to look up functional analysis to figure out hilbert space. Someone told me on this thread to understand Hilbert Space very carefully.

The concept that you really need to understand is "inner product space", which is defined and explained in linear algebra texts. The distinction between "Hilbert space" and "inner product space" is not particularly important. I'll illustrate it with an example (instead of giving a full explanation, which would take some time). ℚ and ℝ (the set of rational numbers, and the set of real numbers) can both be given the structure of an inner product space. This would make ℝ a Hilbert space, but not ℚ. The problem is that there are sequences in ℚ that in a certain sense "look like they should be convergent", but actually aren't. If we view ℚ as a subset of ℝ, we can understand what's going on. Those sequences actually converge to irrational numbers. Nothing like this is possible in ℝ. If a sequence in ℝ "looks like it should be convergent", it actually is. (I.e. it converges to a real number). So we say that ℝ is "complete", and that ℚ isn't. A Hilbert space is a complete inner product space.

As you can probably guess, you won't have to worry about completeness, or even convergence, when you study introductory QM.

 

[Fredrik]

Thank your for putting so much effort into helping me out but I've already decided to go back and learn more of LA. I hope to be back struggling with QM soon.

That's a good idea. I like the book by Axler, because it introduces vector spaces and linear operators as soon as possible, and deals mainly with complex vector spaces. (Avoid books like Anton, which introduce linear operators around page 300, and complex vector spaces even later). Some say that Axler is for people with more mathematical maturity than a typical linear algebra student has. Maybe it is (I can't really tell since I had mathematical maturity when I read it), but studying it will give you some mathematical maturity, so I still think it's the right way to go.

The problem that Ken G suggested is one of the simplest problems in QM, so you might want to take another look at it when you've studied some linear algebra.

As for ODE's and PDE's, I think it it's sufficient to have a thorough understanding of the concepts "function", "derivative" and "partial derivative". So focus on that, and don't worry too much about how to solve differential equations. (Of course, it's certainly not going to hurt to know some of that too, so if you have the time...)

 

[marcusl]

I disagree with your position on diff eqs. A good grounding in PDEs will help the student understand the wave equation (Schrodinger's), solve problems involving separation of variables (such as the hydrogen atom), etc. Especially since the poster is struggling to solve any problems at all, understanding "derivative" as a concept is inadequate. I think PDE's are as important to study as linear algebra adn complex variables.

 

[Fredrik]

But a QM book explains how to solve the differential equations that are relevant to an understanding of QM. I have never understood why people say you need to study differential equations first. (I just searched for the post where someone tried to explain it to me, but I couldn't find it). I agree of course that it will be easier to study QM if you have, just like it would be easier to study differential equations if you have studied QM first.

It seems to me that the he's struggling because he doesn't fully understand essential concepts like orthogonality, and the rest of the items on my list in post #23.