Show that the function
(n and a are constants) is a eigenfunction of the Hamiltonian
operator
. What are the eigenvalues?
and m should be considered constant factors.
**SIDE NOTE: All the question marks should actually be upside down. I did not see the symbol for this!
Show that the function (n and a are constants) is a eigenfunction of the Hamiltonian operator...
9. Show that the function w= sin(x) (n and a are constants) is an eigenfunction of the Hamiltonian operator H = - raxz. What are the eigenvalues? hbar and m should be considered constant factors.
1. Show y = sin ax is not an eigenfunction of the operator d/dx, but is an eigenfunction of the operator da/dx. 2. Show that the function 0 = Aeimo , where i, m, and A are constants, is an eigenfunction of the angular momentum operator is the z-direction: M =; 2i ap' and what are the eigenvalues? 3. Show the the function y = Jź sin MA where n and L are constants, is an eigenfunction of the Hamiltonian...
With the standard Dirac Hamiltonian plus Coulomb potential
below:
a) Show that
.
b) Show that
, where
.
c) Show that
.
d) Since
all mutually commute, they should have common eigenfunctions, and
thus using (c), find the eigenvalues of K2 and K, in
terms of j.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
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.)
Apply the operator
as defined in equation 4.129, and the operator
as defined in equation 4.132 to the hydrogen state
to show that they have the eigenvalues given in
equation 4.133.
[4.129] We were unable to transcribe this image[4.133] We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Expected value and uncertainty
The uncertainty
for the expected value of an observable O is calculated as
The expected value is
of the operator
with a normalized, one-dimensional one Wave function
given by:
a) Show that
b) Show that
, if
is an eigenfunction of the operator
.
We were unable to transcribe this image00= ((50)?) with. ſo = Ô - (Ô). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...
The variational method can be used to solve for the ground state wavefunction and energy of a harmonic oscillator. Using a trail wavefunction of , where the function is defined between . The Hamiltonian operator for a 1D harmonic oscillator is Solving for the wavefunction gives Find that gives the lowest energy and compare from the trial function to the exact value, where coS We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
1. Let be the operator on whose matrix with respect to the standard basis is . a) Verify the result of proof " is normal if and only if for all " for question 1. Note: stands for adjoint b) Verify the result of proof "Orthogonal eigenvectors for normal operators" for question 1. The proof states suppose is normal then eigenvectors of corresponding to distinct eigenvalues are orthogonal. We were unable to transcribe this imageWe were unable to transcribe this...
Solve the system. (Please show all work.) I will be rewriting it
in operator notation as shown below
We were unable to transcribe this imageWe were unable to transcribe this image
Consider the dimensionless harmonic oscillator Hamiltonian,
(where m = h̄ = 1).
Consider the orthogonal wave functions
and
, which are eigenfunctions of H with eigenvalues 1/2 and 5/2,
respectively.
with p=_ïda 2 2 We were unable to transcribe this imageY;(r) = (1-2x2)e-r2/2 (a) Let фо(x-AgVo(x) and φ2(x) = A2V2(x) and suppose that φ。(x) and φ2(x) are normalized. Find the constants Ao and A2. (b) Suppose that, at timet0, the state of the oscillator is given by Find the constant...