Is the function x2e−ax^(2) an eigenfunction of operator d2/dx2 − 4a2x2. If it is then what is the corresponding eigenvalue?
Convince yourself that function exp(-x2/2) is an eigenfunction of the operator (1/2)(-d2/dx2 + x2). Compute the corresponding eigenvalue. (We will see in class that this operator is the Hamiltonian for the harmonic oscillator, if one sets the mass, frequency, and the Planck's constant at 1.)
7. Demonstrate that p -iha/ox is an Hermitian operator. Find the Hermitian conjugate of 7. Demonstrate that p -iha/ox is an Hermitian operator. Find the Hermitian conjugate of
mATRS Assume two linear Hermitian operators A and iB arks (a) What is the adjoint operator of the operator [A, B). Is this again a Hermitian operator? (b) What is the condition that the product of two Hermitian operators is again a Hermitian operator?
An operator A is Hermitian if it satisfies | dx V AV = dr (AW)*v for all v. (a) Show that pl is not Hermitian, where I, k are positive integers. (Hint: First, show that p and are Hermitian. Then show that if A is Hermitian, A" is Hermitian. The remaining piece involves commutators of rand p.] (b) Show that the symmetric combination (c'p + x)/2 is Hermitian.
(a) There are a set of eigenstates ๒n) for the Hermitian operator A with non-degenerate eigenvalues an and a state |ψ Σ¡c; Write down the equation relating the states |>n), the operator A and the eigenvalues a 1. ,n ii. Using Dirac notation explain the requirement for an operator to be Hermitian iii. Explain the relation between the eigenvalues of an operator and the measured iv. For to be properly normalised show the condition required for the values V. Express...
(a) There are a set of eigenstates ๒n) for the Hermitian operator A with non-degenerate eigenvalues an and a state |ψ Σ¡c; Write down the equation relating the states |>n), the operator A and the eigenvalues a 1. ,n ii. Using Dirac notation explain the requirement for an operator to be Hermitian iii. Explain the relation between the eigenvalues of an operator and the measured iv. For to be properly normalised show the condition required for the values V. Express...
(a) Given that the energy operator is Hermitian, explain why hydro- gen atom 2s and 2p, orbitals must be orthogonal. (a) Given that the energy operator is Hermitian, explain why hydro- gen atom 2s and 2p, orbitals must be orthogonal.
qm 09.3 3. An operator  is Hermitian if it satisfies the condition $ $(y) dx = (Ap) u dx, for any wavefunctions $(x) and y(x). (i) The time dependent Schrödinger equation is ih au = fu, at where the Hamiltonian operator is Hermitian. Show that the equation of mo- tion for the expectation value of any Hermitian operator  is given by d(A) IH, Â]), dt ħi i = where the operator  does not depend explicitly on time....
6.5. Prove that the operator L op ? ?? is Hermitian. Suggestion: Follow the procedure outlined in Example 5.2. Keep in mind that the wave function must be single valued