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348-problem15 - ordinates we say that it is separable and...

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Chem. 540 Instructor: Nancy Makri PROBLEM 15 Consider a particle moving on the xy plane. Suppose that the motion of the particle on the plane can be described by a Hamiltonian 1 2 ˆ ˆ ˆ H H H = + where the operators 1 ˆ H and 2 ˆ H describe the motion of a particle in the x and in the y direction, re- spectively: 2 1 1 ˆ ˆ ˆ ( ) 2 x p H V x m = + , 2 2 2 ˆ ˆ ˆ ( ) 2 y p H V y m = + . Suppose we call ( ) n x φ the eigenfunctions of 1 ˆ H with eigenvalues n ε and ( ) k y ψ the eigenfunc- tions of 2 ˆ H with eigenvalues k ξ . Show that the eigenfunction of the total Hamiltonian ˆ H are products of the type ( ) ( ) n k x y φ ψ . Find the corresponding eigenvalues. Whenever the Hamiltonian can be decomposed into a sum of operators involving different co-
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Unformatted text preview: ordinates, we say that it is separable, and as you see its solution is trivial in those cases. A separ -able Hamiltonian means that the motion of the particle in each coordinate is independent of the mo-tion in the other coordinates. This is not the case in general; any potential field that depends simul -taneously on x and on y would make the x motion coupled to the y motion; in that case, the eigen-states would no longer be expressible in product form....
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  • Fall '08
  • Mccall
  • Eigenvalue, eigenvector and eigenspace, Fundamental physics concepts, Eigenfunction, Nancy Makri, able Hamiltonian means

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