Question

1. Consider the vector space of functions r(t) that are solutions of the differential equation d^2r/dt^2 - 6dr/dt + 9r = 0. (i) Show that the vectors: |x1? = e^3t and |x2? = te^3t form a basis for this space. (ii) What is the matrix representation of the operator D^ = d/dt with respect to this basis? (iii) Show that D^ is invertible and calculate the matrix representation of D^-1. (iv) To what mathematical operation does D^-1 correspond? Apply this operation to the basis kets and verify your matrix representation of D^-1. (v) Calculate the two eigenvalues of the matrix D, and show that they are degenerate. (vi) Calculate the corresponding eigenvectors. You should find only one linearly independent eigenvector. What is this eigenvector, expressed as a function? (vii) Can the matrix D be diagonalized? Is this consistent with the diagonalization theorem of Hermitian operators?

          1. Consider the vector space of functions r(t) that are solutions of the differential equation
d^2r/dt^2 - 6dr/dt + 9r = 0.

(i) Show that the vectors:
|x1? = e^3t and |x2? = te^3t
form a basis for this space.

(ii) What is the matrix representation of the operator D^ = d/dt with respect to this basis?

(iii) Show that D^ is invertible and calculate the matrix representation of D^-1.

(iv) To what mathematical operation does D^-1 correspond? Apply this operation to the basis kets and verify your matrix representation of D^-1.

(v) Calculate the two eigenvalues of the matrix D, and show that they are degenerate.

(vi) Calculate the corresponding eigenvectors. You should find only one linearly independent eigenvector. What is this eigenvector, expressed as a function?

(vii) Can the matrix D be diagonalized? Is this consistent with the diagonalization theorem of Hermitian operators?

1. Consider the vector space of functions r(t) that are solutions of the differential equation
d^2r/dt^2 - 6dr/dt + 9r = 0.

(i) Show that the vectors:
|x1? = e^3t and |x2? = te^3t
form a basis for this space.

(ii) What is the matrix representation of the operator D^ = d/dt with respect to this basis?

(iii) Show that D^ is invertible and calculate the matrix representation of D^-1.

(iv) To what mathematical operation does D^-1 correspond? Apply this operation to the basis kets and verify your matrix representation of D^-1.

(v) Calculate the two eigenvalues of the matrix D, and show that they are degenerate.

(vi) Calculate the corresponding eigenvectors. You should find only one linearly independent eigenvector. What is this eigenvector, expressed as a function?

(vii) Can the matrix D be diagonalized? Is this consistent with the diagonalization theorem of Hermitian operators?

Added by Taylor B.

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