What Determines the Independent Variable in Different Quantum Representations?

[dreamsfly]

What's the relationship between an certain represetation and the independent variable under this represetation?
for details:
1 Must the independent variable of X-represetation be X,and P-represetation must be p?if so ,What's the represetation which basic fuctions are P-latent fuctions described with coordinate?
2 if 1 is right,then the state fuctions under E-represetation should contain the independent variable H,L^2,Lz?

Thanks!

 

[Physics Monkey]

Hi dreamsfly,

I'm not sure what exactly you're asking because you're using somewhat unusual terminology, but let me take a crack at it. Simply put, a representation is labeled by the eigenvalues of some set of operators which all commute with each other.

The first representation you learn is called the x-representation, and in this representation the position operator is diagonal. In other words, the position operator X defined on the Hilbert space is represented simply as multiplication by the number x. Wavefunctions are functions of position. On the other hard, the momentum operator is represented in a complicated way in terms of a derivative in the x-representation. The basis states in the x-representation are states of definite position, although as your physical intuition tells you, this notion is somewhat contrived.

Alternatively, you can use the p-representation in which the momentum operator P is represented not as a derivative, but as multiplication by the number p. In this representation the position operator is now represented as a derivative with respect to p, and all wavefunctions are functions of p. As you may know, the x and p representations are connected by the Fourier transform. The basis states in this representation are states of definite momentum (again, these "states" are somewhat unphysical).

There are many different representations you could choose, but one of the most useful is the energy representation, that is, the representation in which the Hamiltonian is diagonal. Here your states are labeled by their energy. If the Hamiltonian is rotationally invariant, these states may also be labeled by definite values of L^2 and L_z. The basic example here would be the energy states of the Hydrogen atom labeled by n (energy eigenvalue E_n = - 13.6/n^2 eV), ell (L^2 eigenvalue ell(ell+1)hbar^2, and m (L_z eigenvalue m hbar).

Hope this helps!

 

[dreamsfly]

Thank you Physics Monkey!it helps a lot!but I still have several questions:
1,I underatand what you mean by "labeled by x or p",but in some books it says themain idea of a representation is the choosing of basic functions which is in order to expand other state functions,here comes the question:if we use the eigenfunctions of p which described by position coordinate to expand the state functions then what is that called?
2,Could E,L^2,L_z label the state synchronously?That is to say they appare in a state function synchronously as the independent variable?
Thanks!

 

[Physics Monkey]

Hi dreamsfly,

1) If you're using the position coordinate as your variable, then you're still using the x-representation. However, you can certainly write down the eigenfunctions (or basis states) of momentum in the position basis. They are simple the plane wave states: exp(i p x / habr). Again, these are states of definite momentum but written in the position basis. If you write your wavefunction in the position basis psi(x) as a superpostion (integral) of the these plane wave states, then the coeffecients of the expansion are the values of the wavefunction in the momentum basis. This is, of course, just the Fourier transform: phi(x) ~ int dp phi(p) exp(i p x / hbar), and in this expansion phi(p) is the wavefunction in the momentum representation. You also recognize the basis functions of the momentum exp(i p x / hbar) written in the position representation.

2) Yes, E, L^2, and L_z can label states simultaneously (provided the Hamiltonian is spherically symmetric). The example I gave above is a good one. The Hydrogen energy states psi_nlm are examples of states which are labeled by energy, total angular momentum, and z component of angular momentum. Of course, the usual wavefunctions, while being labeled by E, L^2, and L_z, are written in the position basis and therefore depend on x.

 

[dreamsfly]

Of course, the usual wavefunctions, while being labeled by E, L^2, and L_z, are written in the position basis and therefore depend on x.

Then is it the X-representation?

 

[dreamsfly]

Ok, I think I have understood it,Thanks again Physics Monkey!