Commutation relations and the uncertainty principle: normalized wave function

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Consider two hermitian operators A and B which satisfy the following commutation relation:

[A, B] = AB-BA=iC, where C is also a hermitian operator in general. Let us introduce a new operator Q defined by: Q=A+ iλB, with λ being a real number, and consider the following scalar product:

Where is any normalized wave function?

(a) Show that Eq. (1) leads to the following result:
(b) Define the uncertainties as follows: With U≡A - <A> and V≡B - < B>, respectively. Show that [U, V] = [A, B] = iC
(c) Thus, replacing A, B in (a) by U,V, we obtain: Regarding I(λ) as a quadratic function ofλ, show that I(λ) is minimum when
(d) Let A=X, B= px, show that the quantum condition [x, px]= i leads to the uncertainty principle:

Estimate the ground state energy of a particle of mass m moving in the potential
V(x) = lambda *(x)^4
by two different methods.
a. Using the Heisenberg Uncertainty Principle;
b. Using the trial function
psi(x)=N*e^{[- abs(x)]/(2a)}
where a is determined by minimizing (E)
*Note abs = absolute value

Consider a particle described by the Cartesian coordinates (x,y,z) = X andtheir conjugate momenta (px, py, pz) = p. The classical definition of the orbital angular momentum of such a particle about the origin is L = X x p.
Let us assume that the operators (Lx, Ly, Lz) = L which represent the components of orbital angular mom

1. Thewavelength spectrum of the radiation energy emitted from a system in thermal equilibrium is observes to have a maximum value which decreases with increasing temperature. Outline briefly the significance of this observation for quantum physics.
2. The “stopping potential” in a photoelectric cell depends only on the f

I am given three unnormalizedwavefunctions for the system:
psi(x) = 100e^x for x<-4
psi(x) = 0.73 cos[(pi)x/40] for-44
I need to determine the probability of thewavefunction vs. x for this system from x=-10 to x=10 so that I can plot it. I have to comment on the probability of fin

Problem 1. Consider a square well that extends from 0 to L.
(a) Write down the general solution for thewavefunction inside the well.
(b) Determine the specific solutions inside the well for the ground state and for the rst excited state by applying the boundary conditions at x = 0 and at x = L.
Now consider a 50:50 superpos

Suppose the minimum uncertainty in the position of a particle is equal to its de Broglie wavelength. If the particle has an average speed of 4.5x10^5 m/s, what is the minimum uncertainty in its speed? Units are in m/s.

** Please see the attached file for the full problem description **
Use Gaussian multiplication on the Hermite polynomials in the attached document. These give the un-normalizedwavefunctions for the levels of the harmonic oscillator.
The 3rd and 4th Hermite polynomials are, respectively:
H_2(x) = 4x^2 - 2
H_3(x) = 8x^

(a) Estimate theuncertainty in the momentum of an electron whose location is uncertain by a distance of 2 Angstrom. What is theuncertainty in the momentum of a proton under the same conditions?
(b) What can one conclude about the relative velocities and energies of the electron and proton in the last problem? Are wave phenom

For a particle in a 1-dimensional box confined between 0the initial state of a particle is given by
phi = phi_1 + 3phi_2 + 2phi_3 (all phi's are functions of x).
a) Normalize this wavefunction.
b) If no energy measurements are made what is the expectation value of energy of this state at a later time t?
c)