Simple question about matrix mechanics

In summary, John is trying to understand how the matrix in the second equation works, but he gets confused because the equation is reversed and the product is not equal to the operator.
  • #1
Jdraper
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HI, I've been running through my lectures notes and have stumbled upon something i can't quite figure out.

I am given

Ψ(x)=∑a_iΨ_i(x)

Then

OΨ(x)=∑ a_i O Ψ_i(x) , where O is an operator acting upon Ψ

Then i am given something which i don't quite understand,

OΨ_i(x) = ∑ O_ji Ψ_j(x) , Where O_ji (i assume) is now a matrix

I understand why the a_i terms disappear in this second equation but I'm unsure why the operator turns into a matrix and why the sum is now over all j's rather than i's

Thanks in advance for your help, John.
 
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  • #2
Well, in QM, we say that for a complete set of eigenstates we have [itex] I=\sum_j \psi_j^\dagger \psi_j [/itex](where I is the identity matrix and [itex] \psi^\dagger [/itex] means complex conjugating the elements of the column matrix and also transposing it). So we can write:
[itex]
O \psi_i=O I \psi_i=O(\sum_j \psi_j^\dagger \psi_j) \psi_i=\sum_j \psi_j^\dagger O \psi_i \psi_j
[/itex]
Now if we set [itex] O_{ji}=\psi_j^\dagger O \psi_i [/itex], we'll have the desired result.
I should add the explanation that the [itex] \psi[/itex]s are column matrices and O was a matrix all along the way. The only difference is, when we write O without subscripts, it means we don't know(or don't write) O's elements and only know what O does to each column matrix. But after a certain point, we find out(or decide to write) O's elements.
 
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  • #3
Ok I think i understand, I am assuming it is a property of the identity matrix that you can input it between O and Ψ and it will still be equal to OΨ. If that is true then i understand everything, think i may brush up on my knowledge of matricies before progressing any further.

Thanks Shyan.
 
  • #4
Oh god...sorry man. I was wrong!
At first, [itex] I=\sum_j \psi_j \psi_j^\dagger [/itex]. The reversed product gives a number and can't be equal to an operator!
Second, I couldn't just send O through the column matrix!
So I should've written:
[itex]
O \psi_i=IO\psi_i=(\sum_j \psi_j \psi_j^\dagger) O \psi_i=\sum_j \psi_j \psi_j^\dagger O \psi_i=\sum_j O_{ji}\psi_j
[/itex]
 
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1. What is matrix mechanics?

Matrix mechanics is a branch of quantum mechanics that uses mathematical matrices to represent the physical properties of quantum systems. It was developed by Werner Heisenberg and Max Born in the 1920s.

2. How does matrix mechanics differ from other approaches to quantum mechanics?

Matrix mechanics differs from other approaches, such as wave mechanics or Schrödinger's equation, in that it does not use the concept of a wave function to describe quantum systems. Instead, it uses mathematical matrices to represent the observable properties of a system.

3. What are the key principles of matrix mechanics?

The key principles of matrix mechanics include the uncertainty principle, which states that the position and momentum of a particle cannot be known simultaneously with certainty, and the principle of superposition, which states that a quantum system can exist in multiple states at the same time.

4. How is matrix mechanics used in practical applications?

Matrix mechanics is used to calculate the probabilities of different outcomes in quantum systems, such as the energy levels of atoms or the behavior of subatomic particles. It is also used in the development of technologies such as quantum computers and sensors.

5. What are the limitations of matrix mechanics?

Matrix mechanics has certain limitations, such as its inability to accurately predict the behavior of systems with more than two particles. It also does not account for the effects of gravity and other forces, which are important in larger scale systems.

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