How Do You Find Negative Eigenvalues in Quantum Mechanics?

[skpang82]

Hi.

In general, how does one find the eigenvalues of an operator?


I have a problem which goes like this:

Consider a 1-D Hamilton operator of the form:

H = (P^2)/2M - |v> V <v| ,​

where the potential strength V is a positive constant and |v><v| is a normalised projector, <v|v> = 1. Determine all negative eigenvalues of H if |v> has the position wave function: <x|v> = sqrt(K) exp(-K|x|) with K>0.

Can someone suggest a rough approach to solving this type of questions? I don't have a clue as to how i should start.

Thanks.

 

[Nate810]

The first step would be to find the eigenvalues and eigenvectors of the operator. To do this, you can use the equation H|ψ> = E|ψ>, where |ψ> is an eigenvector and E is an eigenvalue. You can rearrange the equation to get (H - EI)|ψ> = 0, where I is the identity matrix. This equation can then be written as a matrix equation in terms of the components of |ψ> and the matrix elements of H, which can be solved to obtain the eigenvalues and eigenvectors. Once you have the eigenvalues and eigenvectors, you can determine the negative eigenvalues by simply finding the eigenvalues that are less than zero.

 

[thepatientmental]

Hello, thank you for your question about quantum mechanics. Finding the eigenvalues of an operator is an important aspect of quantum mechanics. In general, the eigenvalues of an operator can be found by solving the eigenvalue equation: H|ψ> = E|ψ>, where H is the operator, |ψ> is the eigenvector, and E is the corresponding eigenvalue. This equation essentially means that when the operator acts on the eigenvector, the resulting vector is a multiple of the original eigenvector, represented by the eigenvalue E. The process for finding the eigenvalues will depend on the specific operator and its properties, but some common methods include diagonalization, perturbation theory, and numerical methods.

As for the specific problem you mentioned, a possible approach could be to first rewrite the Hamiltonian in terms of the position operator, x, and the momentum operator, p. Then, using the given position wave function, you can determine the potential term in the Hamiltonian. From there, you can solve the eigenvalue equation for the remaining kinetic energy term and determine the possible negative eigenvalues. This is just one possible approach, and there may be other methods depending on the context and specific techniques you have learned in your studies. I recommend discussing with your instructor or classmates for additional help and guidance. Good luck with your problem!