What properties determine whether a set of amplitudes forms a vector space?

[spaghetti3451]

Homework Statement

What properties cause complete sets of amplitudes to constitute the elements of a vector space?

Homework Equations

The Attempt at a Solution

Does the question mean 'a vector space' or 'a linear vector space'?

 

[Fredrik]

Homework Statement

What properties cause complete sets of amplitudes to constitute the elements of a vector space?

Homework Equations

The Attempt at a Solution

Does the question mean 'a vector space' or 'a linear vector space'?

A linear vector space and a vector space are the same thing.

I still don't understand the question in the problem. What do they mean by "amplitudes" here? I could guess that they mean complex numbers with absolute value 1 (I'm thinking of probability amplitudes in QM), but then I don't understand the part about "properties" or "complete sets".

 

[spaghetti3451]

A linear vector space and a vector space are the same thing.

Why is there no such structure as a non-linear vector space? Can't we generalise the linearity properties of a linear vector space to obtain a non-linear vector space?

I still don't understand the question in the problem. What do they mean by "amplitudes" here? I could guess that they mean complex numbers with absolute value 1 (I'm thinking of probability amplitudes in QM), but then I don't understand the part about "properties" or "complete sets".

"Amplitudes" means probability amplitudes.

A complete set of probability amplitudes is sufficient to specify the state of a quantum system.

The probability amplitudes form the elements of a vector space because

1. any of the two amplitudes, when added together, results in a third amplitude which is already in the set.
2. any of the amplitudes, when multiplied by a scalar, results in a third amplitude which is already in the set.

I wrote down my answer based on my rudimentary knowledge of vector spaces as applied to quantum mechanics, so I am not confident that my answers are correct.

Furthermore, I don't think the 'third amplitude' I refer to should not be in the complete set of amplitudes because the set is minimal. This is another reason why I think my answers are wrong.

Comments are welcome.

 

[George Jones]

Probability amplitudes, i.e., wave functions, are solutions of the Schrödinger equation. The Schrödinger equation is a linear differential equation, and, as such, its solutions space (for any particular potential) forms a vector space.

 

[Fredrik]

Why is there no such structure as a non-linear vector space? Can't we generalise the linearity properties of a linear vector space to obtain a non-linear vector space?

The most important property of a vector space is that there's an addition operation and a scalar multiplication operation defined on it. This ensures that it's "closed under linear combinations" in the sense that if a,b are scalars and x,y are vectors, ax+by is a vector. Is this the property that you want to drop? These operations are also assumed (by definition of the term "vector space") to satisfy eight conditions. Which ones of those would you like to drop?

"Amplitudes" means probability amplitudes.

A complete set of probability amplitudes is sufficient to specify the state of a quantum system.

The probability amplitudes form the elements of a vector space because

1. any of the two amplitudes, when added together, results in a third amplitude which is already in the set.
2. any of the amplitudes, when multiplied by a scalar, results in a third amplitude which is already in the set.

I wrote down my answer based on my rudimentary knowledge of vector spaces as applied to quantum mechanics, so I am not confident that my answers are correct.

Furthermore, I don't think the 'third amplitude' I refer to should not be in the complete set of amplitudes because the set is minimal. This is another reason why I think my answers are wrong.

The term "complete set of amplitudes" is definitely not standard. I'm not convinced that it makes sense. I'm also not sure how you want to define "probability amplitude". An inner product <x,y> can sometimes be interpreted as the probability amplitude that the measurement will leave the system in state x, given that it was in state y before the measurement. So what would a "complete set" of probability amplitudes be? Since y can be expressed using an orthonormal basis $\{e_i\}$ as $y=\sum_i y_i e_i$, I suppose the set $\{\langle x,y_ie_i\rangle\}_{i=1}^\infty$ could be considered a complete set of amplitudes for a specific problem. But this set is certainly not closed under addition and scalar multiplication.

I think if you want to argue that probability amplitudes form a vector space, you will have to be a lot more specific about what you actually mean by "probability amplitude", "complete set", etc., and maybe also about how you want to define the addition and scalar multiplication operations.

 

[George Jones]

I'm also not sure how you want to define "probability amplitude".

"probability amplitude" is a fairly standard synonym for "wave function" or "quantum state vector".

 

[Fredrik]

"probability amplitude" is a fairly standard synonym for "wave function" or "quantum state vector".

I thought it was a fairly standard synonym for "complex number whose squared absolute value is a probability".

 

[Fredrik]

I thought it was a fairly standard synonym for "complex number whose squared absolute value is a probability".

This is based on, for example, Feynman's "QED: The strange theory of light and matter", where he associates an amplitude with each path and add them up (and normalize) to get the amplitude for detection at the endpoint of the paths. The square of the absolute value of that amplitude is the probability of detection at the endpoint.

So a wavefunction isn't an amplitude, but if ψ is a wavefunction and r is a small but positive real number, then $\sqrt{2r}\,\psi(x)$ is the approximate amplitude that a detector covering the interval [x-r,x+r] will detect the particle.

(I was a bit confused when I talked about amplitudes having absolute value 1 in post #2. That doesn't make much sense).

 

[spaghetti3451]

The most important property of a vector space is that there's an addition operation and a scalar multiplication operation defined on it. This ensures that it's "closed under linear combinations" in the sense that if a,b are scalars and x,y are vectors, ax+by is a vector. Is this the property that you want to drop? These operations are also assumed (by definition of the term "vector space") to satisfy eight conditions. Which ones of those would you like to drop?

Thanks! Now, I have some idea of the restrictions that define a vector space. :)

The term "complete set of amplitudes" is definitely not standard. I'm not convinced that it makes sense. I'm also not sure how you want to define "probability amplitude". An inner product <x,y> can sometimes be interpreted as the probability amplitude that the measurement will leave the system in state x, given that it was in state y before the measurement. So what would a "complete set" of probability amplitudes be? Since y can be expressed using an orthonormal basis $\{e_i\}$ as $y=\sum_i y_i e_i$, I suppose the set $\{\langle x,y_ie_i\rangle\}_{i=1}^\infty$ could be considered a complete set of amplitudes for a specific problem. But this set is certainly not closed under addition and scalar multiplication.

I think if you want to argue that probability amplitudes form a vector space, you will have to be a lot more specific about what you actually mean by "probability amplitude", "complete set", etc., and maybe also about how you want to define the addition and scalar multiplication operations.

What you're saying goes over my head. I took this problem from the Oxford University Professor James Binney's 'The Physics of Quantum mechanics' Chapter 1 problems. Here's the link: http://www-thphys.physics.ox.ac.uk/people/JamesBinney/QBhome.htm

I hope that after a few years of study, I'll be able to understand all your arguments better and make comments on them! :)

 

[Fredrik]

OK, I had a quick look at chapter 1. He defines an amplitude as a complex number whose squared absolute value is a probability. What he means by a "complete set of amplitudes" is a ket/state vector/wavefunction. (These terms mean essentially the same thing). So now I think I understand the question. You are supposed to list the properties of kets that are the reason why the set of kets is a vector space.

The answer consists of the things I mentioned in post #5. There's an addition operation and a scalar multiplication operation on the set of kets, so that if x and y are kets and a and b are complex numbers, ax+by is a ket. And these operations satisfy the eight vector space axioms.

It sounds like you should pick up a book on linear algebra and read it on the side. I like Axler for this. There are several other good books on linear algebra, so you don't have to use Axler, but I think you should choose a book that defines vector spaces and introduces linear transformations as early as possible. I think Axler does the latter around page 40, while Anton does it around page 300, so there are huge differences between the books.

 

[spaghetti3451]

OK, I had a quick look at chapter 1. He defines an amplitude as a complex number whose squared absolute value is a probability. What he means by a "complete set of amplitudes" is a ket/state vector/wavefunction. (These terms mean essentially the same thing). So now I think I understand the question. You are supposed to list the properties of kets that are the reason why the set of kets is a vector space.

The answer consists of the things I mentioned in post #5. There's an addition operation and a scalar multiplication operation on the set of kets, so that if x and y are kets and a and b are complex numbers, ax+by is a ket. And these operations satisfy the eight vector space axioms.

I see! So, to reiterate your point, a ket/state vector/wavefunction |ψ> = a₁|1> + a₂|2> + a₃|3> + ... + a_n|n> = (a₁, a₂, a₃, ... , a_n), where the a_n's are eigenvalues (possible outcomes of a measurement on the state of a system with wavefunction ψ) and the |n>'s are the eigenvectors (possible states that a system can collapse into after a measurement performed on the system).

The set (a₁, a₂, a₃, ... , a_n) is the 'complete set of amplitudes', isn't it?

It sounds like you should pick up a book on linear algebra and read it on the side. I like Axler for this. There are several other good books on linear algebra, so you don't have to use Axler, but I think you should choose a book that defines vector spaces and introduces linear transformations as early as possible. I think Axler does the latter around page 40, while Anton does it around page 300, so there are huge differences between the books.

I think you mention this point because my knowledge of vector spaces is rudimentary? Anyway, thanks for the titles. I'll try to read the books of Axler and Anton from cover to cover as they will help me with the mathematics of quantum mechanics and relativity.

 

[Fredrik]

I see! So, to reiterate your point, a ket/state vector/wavefunction |ψ> = a₁|1> + a₂|2> + a₃|3> + ... + a_n|n> = (a₁, a₂, a₃, ... , a_n), where the a_n's are eigenvalues (possible outcomes of a measurement on the state of a system with wavefunction ψ) and the |n>'s are the eigenvectors (possible states that a system can collapse into after a measurement performed on the system).

That sum would usually have infinitely many terms, not stop at some integer n. The notation $(a_1,a_2,\dots)$ should be avoided. Even if the number of components had been finite, it would be more appropriate to represent the vector as a column matrix
$$\begin{pmatrix}a_1\\ a_2\\ \vdots\end{pmatrix}$$ because then a linear operator acting on the vector can be represented as a matrix that multiplies this matrix of components from the left.

The set (a₁, a₂, a₃, ... , a_n) is the 'complete set of amplitudes', isn't it?

Yes, I think that's a valid way of looking at it.

I think you mention this point because my knowledge of vector spaces is rudimentary?

Yes, you didn't seem to be familiar with the definition of "vector space". But there's also the fact that linear algebra is so useful in QM. There's a complex Hilbert space (a special kind of inner product space) associated with each physical system. Preparation procedures (ways the system can be prepared before the measurement begins) are represented by vectors. Measuring devices are represented by self-adjoint linear operators. The possible results correspond to eigenvalues and eigenvectors. Probabilities of measurement results are computed using the inner product. Etc.

Anyway, thanks for the titles. I'll try to read the books of Axler and Anton from cover to cover as they will help me with the mathematics of quantum mechanics and relativity.

I actually didn't mean to recommend Anton, so let me explain what I think about it. It's an excellent book. But, in the 6th edition at least, which is the only one I'm familiar with, the things we really need are postponed to the very end. So the order of the topics is bad for a physics student. That's why I prefer Axler. Axler starts with complex vector spaces right away, and introduces linear operators (linear transformations) as soon as possible. However, since Axler doesn't start with a bunch of "how to calculate" stuff like Anton, and starts proving theorems right away, people who have no experience with proofs seem to find it difficult.

There are many books that are good enough, and it doesn't matter much which one(s) you study. The one by Friedberg, Insel & Spence seems to be good as well, and there are several free books available online.

Here's something I said in another thread, about which topics are important:

Complex vector spaces, linear independence, bases, inner products, inner product spaces, orthonormal bases, linear operators, matrices, matrix multiplication, a theorem about which matrices are invertible, the relationship between linear operators and matrices, the adjoint operation, self-adjoint linear operators, eigenvectors and eigenvalues, and the spectral theorem.

Since the relationship between linear operators and matrices is very important, I recommend that you use a book that presents those things early in the book, like Axler or Friedberg, Insel & Spence. (I have only read the former, but I've heard good things about the latter).

You may not need all of those things for an introductory course. It may be enough to understand complex inner product spaces, orthonormal bases and self-adjoint linear operators. But you will need the rest if you want to get good at QM.