Why do you need infinite size matrix which commute....

In summary: So, too, are the momentum eigenfunctions and energy eigenfunctions.In summary, the concept of linear independence in the context of wavefunctions on the real line means that for any set of wavefunctions, there is no way to add them together to get a function that is identically equal to zero. This is because there are an infinite number of unique wavefunctions on the real line, and they cannot be expressed as a finite combination of other wavefunctions. This is important in quantum mechanics, as it allows for the possibility of infinite-dimensional matrices to represent operators such as position and momentum.
  • #1
Phys12
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...to give a number?

https://ocw.mit.edu/courses/physics...g-2016/lecture-notes/MIT8_04S16_LecNotes5.pdf

On page 6, it says,
"
Matrix mechanics, was worked out in 1925 by Werner Heisenberg and clarified by Max Born and Pascual Jordan. Note that, if we were to write xˆ and pˆ operators in matrix form,they would require infinite dimensional matrices. One can show that there are no finite size matrices that commute to give a number times the identity matrix, as is required from (2.13). This shouldn’t surprise us: on the real line there are an infinite number of linearly independent wavefunctions, and in view of the correspondences in (2.14) it would suggest an infinite number of basis vectors. The relevant matrices must therefore be infinite dimensional.
"

Equation 2.13: [x, p] = ihbar

2.14 Correspondences:
operators ↔ matrices
wavefunctions ↔ vectors
eigenstates ↔ eigenvectors

The part that I don't get is, "on the real line, there are an infnite number of linear independent wavefunctions." Why is it so? If I think of the real line and think of wavefunctions as vectors, every single vector will be linearly *dependent* not independent, right? What am I missing here?
 
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  • #2
Phys12 said:
If I think of the real line and think of wavefunctions as vectors, every single vector will be linearly *dependent* not independent, right?
I think what you may be missing is the notion of what it means for two functions to be equal. Two functions are equal if and only if they have the same domain and, for every element of that domain, they give the same result. So the sine and cosine functions are unequal, even though they intersect infinitely many times on the real numbers (at ##x=(k+0.25)\pi## for any integer ##k##), because for all other inputs they give different results.

So for a set of functions to be dependent, there needs to be some linear combination of them that is the function that gives zero for every input - called the zero function.

The infinite set of functions:
$$\{x\mapsto \sin kx\ :\ k\in\mathbb Z\}$$
is linearly independent because there is no linear combination of them that gives the zero function.
 
  • #3
Consider the usual 3d directions: x, y and z. These are linearly independent because there is no way to add a vector pointing in the x and a vector pointing in the y direction to get a vector pointing in the z direction.

On the real line, one can consider "wave functions" that are infinitely narrow and peaked or located at only a single position - let's call these position eigenfunctions. There is no way to add a position eigenfunction located at x1 and a position eigenfunction located at x2 to get a position eigenfunction located at x3. Thus the position eigenfunctions that are located at different positions are linearly independent.
 
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1. Why do you need an infinite size matrix?

An infinite size matrix is a mathematical tool that allows for the representation of an infinite number of variables or data points. It is useful in fields such as physics, engineering, and computer science where complex systems require a large number of inputs to accurately model and analyze.

2. What does it mean for a matrix to commute?

Two matrices commute if their multiplication is the same regardless of the order in which they are multiplied. In other words, the order of multiplication does not affect the result. This property is important in linear algebra and is used in many applications, such as solving systems of equations.

3. Why is it important for an infinite size matrix to commute?

When dealing with infinite size matrices, it is important for them to commute because it allows for easier manipulation and simplification of equations. This can lead to more efficient and accurate solutions in various mathematical problems.

4. Can an infinite size matrix commute with all other matrices?

No, not all infinite size matrices will commute with all other matrices. It depends on the specific properties and elements of the matrices. However, there are certain types of infinite size matrices, such as diagonal matrices, that will commute with all other matrices.

5. How is an infinite size matrix used in real-world applications?

Infinite size matrices are commonly used in physics and engineering to model and analyze complex systems. They can also be used in computer science, such as in machine learning algorithms, to process and analyze large amounts of data. Additionally, infinite size matrices have applications in economics, chemistry, and other fields where a large number of variables need to be considered.

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