Matrix element (raising and lowering operators)

In summary, you need to know your hamiltonian as a function of the position and momentum operators. Once you know that, you can solve for the eigenstates of the hamiltonian in the position or the momentum representation (by solving the associated Schroedinger equation). Then you need to find a link between your specific given basis of energy eigenstates you mentionned, and the solutions of the Schroedinger equation you found in the (say) position representation. This mapping can be trivial (in the non-degenerate case) because for same eigenvalues, you have a clear identification between your solution of the Schroedinger equation with that eigenvalue and the given energy eigenstate (well, up
  • #1
ayalam
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0
How to determine the matrix representation of position & momentum operator using the energy eigenstates as a basis
 
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  • #2
ayalam said:
How to determine the matrix representation of position & momentum operator using the energy eigenstates as a basis

You need to know your hamiltonian as a function of the position and momentum operators. Once you know that, you can solve for the eigenstates of the hamiltonian in the position or the momentum representation (by solving the associated Schroedinger equation). Then you need to find a link between your specific given basis of energy eigenstates you mentionned, and the solutions of the Schroedinger equation you found in the (say) position representation. This mapping can be trivial (in the non-degenerate case) because for same eigenvalues, you have a clear identification between your solution of the Schroedinger equation with that eigenvalue and the given energy eigenstate (well, up to a phase factor...). It can also be more complicated in the degenerate case. Then you'll need to look at how exactly the energy eigenstates of your given basis are defined.

In any case, you end up by finding an equivalence:

f_E (x) <--> |E> ; so you can consider that f_E(x) = <x|E>

This means that f_E(x), now seen as F(E,x) is the matrix element of the basis transformation that maps the basis {|x>} into the basis {|E>}. That's sufficient to transform your representation of the position operator X (which is simply x delta(x-x0) for <x|X|x0>) into the |E> basis <E |X|E> ; and in the same way to transform the representation of P in the |x> basis into a representation in the |E> basis.

What I've outlined above is the pedestrian method. Sometimes more elegant algebraic methods exist: for instance in the case of the harmonic oscillator, with the creation and annihilation operators. I don't know how general that approach is.

cheers,
Patrick.
 
  • #3
Funny,the title mentioned raising & lowering ladder operators.Could he possibly mean the SHO?:rolleyes:

Daniel.
 
  • #4
yes i did mean sho.
 
  • #5
Voilà.So you're interested in making a unitary transformation from the matrix

[tex] \langle i|\hat{H}|j\rangle [/tex] (i,j can run from 0 to +infinity)

to [tex] \langle i|\hat{x}|j\rangle [/itex] and [tex] \langle i|\hat{p}|j\rangle [/itex] ,where,now

[tex] |i\rangle \longrightarrow \langle x|i\rangle [/itex]

,i.e.u'll be needing the SHO-s wavefunctions.U can use only coordinate ones,just as long as u express the momentum operator in the same basis [itex] \{\langle x| \} [/itex].


Daniel.
 

What is a matrix element?

A matrix element is a numerical value that represents the entry of a specific row and column in a matrix. It is used in linear algebra and quantum mechanics to represent the relationship between two states or operators.

What is the purpose of raising and lowering operators?

Raising and lowering operators are used in quantum mechanics to change the energy state of a particle. The raising operator increases the energy state by one unit, while the lowering operator decreases it by one unit.

How are raising and lowering operators related to each other?

Raising and lowering operators are related through their commutation and anti-commutation relations. These relations dictate the order in which the operators can be applied and the resulting outcomes.

What are the properties of raising and lowering operators?

Raising and lowering operators have several important properties, including linearity, hermiticity, and orthogonality. These properties allow them to be used in mathematical operations and to represent physical properties of quantum systems.

How are raising and lowering operators used in quantum mechanics?

Raising and lowering operators are used to describe the behavior and properties of quantum systems, such as energy levels and transition probabilities. They are also used in the creation and annihilation of particles in quantum field theory.

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