QM: Ladder Operators Explained Step-by-Step

[sillystring]

I am taking a QM course and we are using griffiths intro to QM text, 2nd edition. I like the text but I find it lacking when it comes to explaining ladder operators. I need to see how to use them in a very detailed step-by-step problem. Does anyone know of any good textbooks or websites that explain the ladder operators very well?
Thanks

 

[Mentz114]

Ladder operators ( also called creation and annhiliation operators) operate on Fock space vectors. You can make a toy Fock space and see the action of the ladder operators like this -

Lets say our field has a mode that has 5 energy levels. So the vectors
(1, 0, 0, 0, 0 ), ( 0, 1, 0, 0, 0 ), (0, 0, 1, 0, 0 ), ( 0, 0, 0, 1, 0, ), (0, 0, 0, 0, 1 ) form a basis for the Fock space. The ladder operators look like -

$$\left(\begin{array}{ccccc} 0 &1 &0 &0 &0 \\0 &0 &1 &0 &0\\0 &0 &0 &1 &0 \\0 &0 &0 &0 &1\\1 &0 &0 &0 &0\end{array}\right)$$

The other operators has the 1's below the principal diagonal. Try multiplying an operator into a basis vector.

This may not be be a great analogy, so don't take it too literally.

 

[pmb_phy]

I am taking a QM course and we are using griffiths intro to QM text, 2nd edition. I like the text but I find it lacking when it comes to explaining ladder operators. I need to see how to use them in a very detailed step-by-step problem. Does anyone know of any good textbooks or websites that explain the ladder operators very well?
Thanks

I created a web page at one time for various purposes, one of which was to demonstrate a use of the creation and annihilation (ladder) operators. Please see - http://www.geocities.com/physics_world/qm/harmonic_oscillator.htm - and let me know if it was helpful.

Pete

 

[quetzalcoatl9]

once you have derived the solution to the QHO in terms of the hermite polynomials, you will come to appreciate simplicity and elegance of the ladder operators!

(if only we had ladder operators available to us analytically for more complex systems).

also, since a field can be constructed as a collection of harmonic oscillators, in the limit of quantization the operators become enormously useful and are the basis for much of QFT and QED.

 

[StatMechGuy]

Sakurai does a great job discussing creation and annihilation operators, but if you're using Griffith's abomination then Sakurai may be a bit of a stretch. Shankar also does an okay job of it, but you have to slog through more verbiage than any textbook should ever have to get it.

 

[nrqed]

Ladder operators ( also called creation and annhiliation operators) operate on Fock space vectors. You can make a toy Fock space and see the action of the ladder operators like this -

Lets say our field has a mode that has 5 energy levels. So the vectors
(1, 0, 0, 0, 0 ), ( 0, 1, 0, 0, 0 ), (0, 0, 1, 0, 0 ), ( 0, 0, 0, 1, 0, ), (0, 0, 0, 0, 1 ) form a basis for the Fock space. The ladder operators look like -

$$\left(\begin{array}{ccccc} 0 &1 &0 &0 &0 \\0 &0 &1 &0 &0\\0 &0 &0 &1 &0 \\0 &0 &0 &0 &1\\1 &0 &0 &0 &0\end{array}\right)$$

The other operators has the 1's below the principal diagonal. Try multiplying an operator into a basis vector.

This may not be be a great analogy, so don't take it too literally.

Just to point out a typo: there should be no "1" in the lower left corner fo the matrix. I am sure this is just a typo.

 

[Mentz114]

Hi nrqed,
actually I put in deliberately to make the operator cyclic. The last row takes
the vector (1,0,0,0,0 ) to (0,0,0,0,1), back to the beginning, so to speak.
In real situations the dimension is unbounded so you don't need this trick.

So strictly speaking, you're right, it shouldn't there.

 

[CarlB]

Hi nrqed,
actually I put in deliberately to make the operator cyclic. The last row takes
the vector (1,0,0,0,0 ) to (0,0,0,0,1), back to the beginning, so to speak.
In real situations the dimension is unbounded so you don't need this trick.

So strictly speaking, you're right, it shouldn't there.

This gets to a subject I've been trying to understand recently, perhaps you can help. The subject is possible extensions of the standard mass term in QFT. The usual is

$$m(\psi^\dag_L\psi_R + \psi^\dag_R\psi_L)$$

this corresponds, in your matrix notation, to a cyclic operator:
$$\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$

Now the above is a nice operator because it is Hermitian. What I'd like to know is how one would deal with a mass operator whose matrix form looks like the cyclic matrix you wrote down, which is non Hermitian. Non Hermitian operators are associated with violations of PT symmetry, which is why they are of some interest in elementary particles.

The particle I'd like to apply this to is the neutrino. The idea is to give the neutrino low mass by assuming a bunch of sterile neutrinos. There would be one state, $$\psi_L$$ that suffers the weak interaction, everything else is sterile. All the sterile neutrinos would, of course, be unobservable. My reason for looking at this is to find a way to explain the low mass of the neutrino in the context of density operator formalism.

Carl

 

[quantumdude]

Ladder operators ( also called creation and annhiliation operators) operate on Fock space vectors.

Eh? I thought that ladder operators (namely $J_{\pm}=J_x\pm iJ_y$) operate on angular momentum eigenstates. Or are you alluding to Wigner's oscillator model of angular momentum?

 

[dextercioby]

Eh? I thought that ladder operators (namely $J_{\pm}=J_x\pm iJ_y$) operate on angular momentum eigenstates. Or are you alluding to Wigner's oscillator model of angular momentum?

Nope, "ladder operators" refers to the operators appearing both in the theory of harmonic oscillator and the theory of the hydrogen atom, to name 2 other examples.

 

[Mentz114]

PMB's webpage, (URL above ) gives a very clear exposition of the ladder ops for the SHO. Very nice. A lot easier to follow than Dirac.

 

[sillystring]

I am processing all this information, the post by pmb_phy has proven very helpful so far