Calculating Commutator [H,U(m,n)] with Homework Statement

• Berny
In summary, The commutator [H,U(m,n)] is given by (E(m)-E(n))U(m,n), where E(m) and E(n) are the eigenvalues of the hermitian operator H corresponding to the eigenstates |phi(m)> and |phi(n)>, respectively. This commutator does not depend on the state of the quantum system.
Berny

Homework Statement

|phi (n)> being eigen states of hermitian operator H ( H could be for example the hamiltonian
of anyone physical system ). The states |phi (n)> form an orthonormal discrete basis.
The operator U(m,n) is defined by:

U(m,n)= |phi(m)><phi(n)|
Calculate the commutator:
[H,U(m,n)]

( this is part of the first problem in Cohen, Tannoudji, Diu, Laloe textbook in quantum mechanics.)

The Attempt at a Solution

$\$

[HU-UH] (ψ) = H|phi(m)><phi (n)|ψ> - |phi (m)><phi(n)| H| ψ>

= <phi(n)|ψ> H |phi (m>) - |phi (m><phi(n)| <ψ | H

and then ? i did not find symbol phi.

Note that the operator ##H## can be written as ##H=\sum\limits_{k}E_{k}\left|\phi(k)\right>\left<\phi(k)\right|##, where the ##E_{k}## are its eigenvalues. Also, note that the vectors ##\left|\phi(k)\right>## form an orthonormal set.

you should use an eigenstate of the Hamiltonian instead of $\psi$ (if you ask how you can do that, you can expand $\psi$ as a superposition of the Hamiltonian's eigenstates)
Then in general, following correct paths you will reach the desired result/

Berny said:

Homework Statement

|phi (n)> being eigen states of hermitian operator H ( H could be for example the hamiltonian
of anyone physical system ). The states |phi (n)> form an orthonormal discrete basis.
The operator U(m,n) is defined by:

U(m,n)= |phi(m)><phi(n)|
Calculate the commutator:
[H,U(m,n)]

( this is part of the first problem in Cohen, Tannoudji, Diu, Laloe textbook in quantum mechanics.)

The Attempt at a Solution

$\$

[HU-UH] (ψ) = H|phi(m)><phi (n)|ψ> - |phi (m)><phi(n)| H| ψ>

= <phi(n)|ψ> H |phi (m>) - |phi (m><phi(n)| <ψ | H

and then ? i did not find symbol phi.

many thanks for help. I find a commutator value depending upon the system energy :

commutator = E(m) U(m,n) if E= E(m) and - E(n) U(m,n) if E=E(n).

in other cases it's zero.
Is this correct ?

I got something like [H,U(m,n)] = (E(m)-E(n))U(m,n). The commutator does not depend on what state the quantum system is in.

Berny said:

Homework Statement

|phi (n)> being eigen states of hermitian operator H ( H could be for example the hamiltonian
of anyone physical system ). The states |phi (n)> form an orthonormal discrete basis.
The operator U(m,n) is defined by:

U(m,n)= |phi(m)><phi(n)|
Calculate the commutator:
[H,U(m,n)]

( this is part of the first problem in Cohen, Tannoudji, Diu, Laloe textbook in quantum mechanics.)

The Attempt at a Solution

$\$

[HU-UH] (ψ) = H|phi(m)><phi (n)|ψ> - |phi (m)><phi(n)| H| ψ>

= <phi(n)|ψ> H |phi (m>) - |phi (m><phi(n)| <ψ | H
You can say that ##\hat{H}\lvert \phi_m \rangle \langle \phi_n \vert \psi \rangle = \langle \phi_n \vert \psi \rangle \hat{H}\lvert \phi_m \rangle## because ##\langle \phi_n \vert \psi \rangle## is a number, though it doesn't really help you in this case. What you can't do is say ##\lvert \phi_m \rangle \langle \phi_n \lvert \hat{H} \rvert \psi \rangle## equals ##\lvert \phi_m \rangle \langle \phi_n \lvert \langle \psi \rvert \hat{H}## because ##\hat{H}\lvert \psi \rangle## and ##\langle \psi \rvert \hat{H}## aren't the same. One's a bra; the other, a ket. You need to be a bit more precise with your notation, otherwise you're invariably going to make errors.

You have, so far,
\begin{align*}
[\hat{H},\hat{U}] &= \hat{H}\hat{U} - \hat{U}\hat{H} \\
&= \hat{H}\lvert \phi_m\rangle\langle\phi_n\rvert - \lvert \phi_m\rangle\langle\phi_n\rvert\hat{H}
\end{align*} Now in the first term, apply ##\hat{H}## to the ket ##\lvert \phi_m \rangle##. What do you get? Similarly, in the second term, what do you get when ##\hat{H}## acts on the bra ##\langle \phi_n \rvert##?

What is a commutator?

A commutator is an operation in mathematics and physics that measures how much two mathematical objects fail to commute with each other. In simpler terms, it measures the extent to which the order in which two operations are performed affects the final result.

What is the formula for calculating a commutator?

The formula for calculating a commutator of two operators, A and B, is [A,B] = AB - BA. This means that you multiply A and B in one order and then subtract the result of multiplying them in the opposite order.

What is the significance of calculating a commutator?

Calculating a commutator is important in quantum mechanics, as it helps us understand the properties of quantum systems and how they evolve over time. It also allows us to determine which physical quantities can be measured simultaneously with precision.

How do I calculate the commutator [H,U(m,n)]?

To calculate the commutator [H,U(m,n)], you will need to know the Hamiltonian operator, H, and the operator U(m,n). You can then plug these values into the formula [H,U(m,n)] = HU(m,n) - U(m,n)H and solve for the resulting operator.

What are some real-world applications of calculating a commutator?

Calculating commutators has many applications in physics, such as in quantum mechanics, electromagnetism, and relativity. It is also used in engineering fields, such as in signal processing and control systems.

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