Physical quantities versus wave function?

[wasi-uz-zaman]

hi, please explain can by using suitable operator we can find any physical quantity- as by using hamiltonian on wave function we can find energies by the eigenvalues?
thanks
wasi-uz-zaman

 

[wotanub]

The classical physical quantity only arises if the state is an eigenstate of the operator.

Just like in linear algebra, multiplying a matrix by one of its eigenvectors has the same effect of multiplying the eigenvector by a constant (the corresponding eigenvalue).

In your example, the time independent Schrödinger equation, the wavefunction is an eigenfunction of the Hamiltonian, or some linear combination of eigenfunctions.

 

[wasi-uz-zaman]

does it mean operator can only apply to eigenfunction?

 

[wotanub]

No, but it's difficult to determine the action of an operator on something that isn't an eigenfunction, so we always try to write wave functions as a superposition of eigenfunctions of an operator to determine its action.

 

[Couchyam]

Not all physical quantities are eigenvalues of some operator, time being the most immediate example.
When a quantum system is described by a wave function, the time evolution is determined by a unitary operator that acts on the Hilbert space used to describe the particle. If the Hilbert space has dimension "n", then the time evolution operator over a given time interval is an element of U(n). Now, physics enters because we do not necessarily have the capacity to reproduce the effect of an arbitrary element of U(n). A physical interaction will have the effect of causing the system to move within some continuous subgroup S of U(n). An interaction over an infinitesimal interval of time gives rise to an element of the Lie algebra "s" associated with S, which is a vector in the tangent space of the identity of S. By our assumptions, this "s" can be used to build physical Hamiltonian operators, and acts on the Hilbert space in a way that is consistent with the action of S.
The Hilbert space used to describe the wave function decomposes into eigenspaces under the action of 's', and these eigenspaces are 'distinguishable' because they generally have different dynamical properties (consider the differences between singlet and triplet states in a two-electron system: one exhibits entanglement in all reference frames, while the other does not). The actual eigenvalues of 's' can be determined in the finite-dimensional case through verifying certain global symmetries of the system, or through measuring the density of states in an experimental setup where the microcanonical ensemble Hamiltonian is given by 's'.