Matrix representation of ladder operators

In summary, the task is to find the matrices representing the ladder operators a+, a-, and a+a- on Hilbert space, using the equations a+=1/sqrt(2hmw)*(-ip+mwx), a_=1/sqrt(2hmw)*(ip+mwx), and a+a_= 1/hw*(H) - 1/2. This can be done by first finding the matrix for a+ and then taking the conjugate of its transpose to get the matrix for a_. Multiplying these two matrices will give the matrix for a+a-. The equations for the ladder operators are also given in Griffiths Quantum Mechanics text (equation 4.120) and can be used to find the matrix representations
  • #1
Ed Quanta
297
0

Homework Statement



Find the matrices which represent the following ladder operators a+,a_, and a+a-
All of these operators are supposed to operate on Hilbert space, and be represented by m*n matrices.

Homework Equations



a+=1/square root(2hmw)*(-ip+mwx)
a_=1/square root(2hmw)*(ip+mwx)
a+a_= 1/hw*(H) - 1/2


The Attempt at a Solution



I recognize that once I find the matrix which represents a+, a_ will be represented by the conjugate of the transpose of that matrix. I also can then find a+a_ by matrix multiplication. Not sure about anything else though.
 
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  • #2
Do you know the action of the ladder operators onto the standard basis vectors in the state space ?
 
  • #3
No clue. There is nothing on this in my Griffiths quantum mechanics text that I have seen.
 
  • #4
Ed Quanta said:
No clue. There is nothing on this in my Griffiths quantum mechanics text that I have seen.

Griffiths does give the equation. It is 4.120 and 4,121 (given in problem 4.18), Second edition.

[tex] L_{\pm} |l, m> = \hbar {\sqrt{l(l+1)-m(m \pm 1)}}|l,m\pm 1> [/tex]
 
  • #5
But we haven't gotten to chapter 4 yet, or done anything having to do with angular momentum. The ladder operators were introduced with respect to the harmonic oscillator.

Would I still just plug in values for l and m to find the elements of the matrix?

We were told that
(a+)mn=∫ψn(x)(a+)(ψm(x))dx

And same type of definition for (a_)mn only with a_ instead of a+.

Is there any way for me to come up with the equation in Griffiths 4.120 given this information?

I'm sorry if I am slow.
 
Last edited:
  • #6
Ed Quanta said:
But we haven't gotten to chapter 4 yet, or done anything having to do with angular momentum. The ladder operators were introduced with respect to the harmonic oscillator.

Would I still just plug in values for l and m to find the elements of the matrix?

We were told that
(a+)mn=∫ψn(x)(a+)(ψm(x))dx

And same type of definition for (a_)mn only with a_ instead of a+.

Is there any way for me to come up with the equation in Griffiths 4.120 given this information?

I'm sorry if I am slow.

My APOLOGIES! I was thinking about the ladder operators for angular momenta (I read replies to your posts but not your initial post in details...my bad!:redface: :redface: :frown: :cry: )

My sincerest apologies.

Ok, what you need then is his equation 2.66, page 48 (in the second edition):

[tex] a_+ \psi_n = {\sqrt{n+1}} \psi_{n+1} \,\,\,\,\,\, a_- \psi_n = {\sqrt{n}} \psi_{n-1} [/tex]

If the [itex]\psi_n[/itex] are represented by explicit column vectors (usually the ones with 1 in one slot and zero and the others), it is easy to use that to find the matrix representations of the ladder operators (they will be matrices with entries below or above the diagonal only)

My apologies, again.

Patrick
 
  • #7
Thanks agaom. Easier than I thought. That is what I got on wikipedia but I wasn't sure why until now.
 

1. What is the purpose of using matrix representation for ladder operators?

The matrix representation of ladder operators is used to simplify and solve problems in quantum mechanics. It allows us to represent complex operators as matrices, making it easier to perform calculations and analyze the behavior of quantum systems.

2. How are the ladder operators represented in matrix form?

The ladder operators, or creation and annihilation operators, are represented as matrices with specific properties. The creation operator is represented as a lower triangular matrix, while the annihilation operator is represented as an upper triangular matrix. Both matrices have zeros on the diagonal and follow a specific pattern depending on the dimensions of the system.

3. What is the relationship between the ladder operators and the eigenstates of a system?

The ladder operators are closely related to the eigenstates of a system. The eigenstates are the states of a system that are associated with definite energy values. The ladder operators act on these eigenstates to change the energy level of the system. The creation operator increases the energy by one unit, while the annihilation operator decreases it by one unit.

4. How do the ladder operators commute with the Hamiltonian operator?

The ladder operators commute with the Hamiltonian operator, meaning that they share common eigenstates. This property is crucial in quantum mechanics as it allows us to determine the energy levels of a system and understand its dynamics. The commutation relationships between the ladder operators and the Hamiltonian are also used to derive important equations, such as the Heisenberg uncertainty principle.

5. Can the matrix representation of ladder operators be used for any quantum system?

Yes, the matrix representation of ladder operators can be used for any quantum system, regardless of its dimensions. The matrices representing the ladder operators will have different dimensions depending on the system, but they will still follow the same rules and properties. This makes the matrix representation a versatile tool in quantum mechanics for solving a wide range of problems.

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