Chapter 5, Problem 21Q

A certain atom remains in an excited state for about 51.7 ns before emitting a 2.15-eV photon and transitioning to the ground state. What is the uncertainty in the frequency of the photon in Hz?

A nucleus emits a gamma ray of energy 1.2 MeV from a state that has a lifetime of 2.1 ns. What is the uncertainty in the energy of the gamma ray? The best gamma-ray detectors can measure gamma-ray energies to a precision of no better than a few eV. Will this uncertainty be directly measurable?

For a quantum particle of mass m in the ground state of a square well with length L and infinitely high walls, the uncertainty in position is Δx ≈ L. (a) Use the uncertainty principle to estimate the uncertainty in its momentum.(b) Because the particle stays inside the box, its average momentum must be zero. Its average squared momentum is then ⟨p2⟩ ≈ (Δp)2. Estimate the energy of the particle. (c) State how the result of part (b) compares with the actual ground-state energy.

Angular momentum in quantum mechanics is given by L = Lxi+Lyj+Lzk with components Lx = ypz- zpy, Ly = zpx - xpz, Lz = xpy - ypx. a) Use the known commutation rules for x, y, z, px, py and pz to show that [Ly, Lz] = ihLx. b) Consider the spherical harmonic Y1, -1([theta], [phi]) = (1/2)*sqrt(3/2pi)*sin[theta]*e-i[phi], where [theta] and [phi] are the polar and azimuthal angles, respectively. -> i) Express Y1, -1 in terms of cartesian coordinates.  -> ii) Show that Y1, -1 is an eigenfunction of Lz. ci) Express the wavefunction [psi]210 for the 2pz orbital of the hydrogen atom (derived in the lectures and given in the notes) in cartesian coordinates. [Note: This involves a different spherical harmonic than in (b).] ii) Based on this expression, show that this wavefunction satisfies the three-dimensional stationary Schrodinger equation of the hydrogen atom, and determine the corresponding energy.   I have attached the question better formatted, as well as the information from…

A scientist claims that the position of an electron in an atom is measured to an accuracy of 0.000100 pm. What is the electron’s uncertainty in velocity?

Imagine an alternate universe where the value of the Planck constant is 6.62607x10−17J·s. In that universe, which of the following objects would require quantum mechanics to describe, that is, would show both particle and wave properties? Which objects would act like everyday objects, and be adequately described by classical mechanics? A bacterium with a mass of 9.0 pg, 6.0 µm long, moving at 9.00 µm/s. A mosquito with a mass of 2.3 mg, 6.0 mm long, moving at 3.0 m/s. A paper airplane with a mass of 5.9 g, 295. mm long, moving at 3.7 m/s. A car with a mass of 2000. kg, 4.4 m long, moving at 81.0 km/h.

Imagine another universe in which the value of Planck’s constant is 0.0663 J . s, but in which the physical laws and all other physical constants are the same as in our universe. In this universe, two physics students are playing catch. They are 12 m apart, and one throws a 0.25 kg ball directly toward the other with a speed of 6.0 m/s. (a) What is the uncertainty in the ball’s horizontal momentum, in a direction perpendicular to that in which it is being thrown, if the student throwing the ball knows that it is located within a cube with volume 125 cm3 at the time she throws it? (b) By what horizontal distance could the ball miss the second student?

In 1927, Heisenberg proposed his famous Uncertainty Principle that extended the conceptual grounds of quantum theory in surprising ways. In short, Heisenberg stated that: O In principle, no two atomic particles in an orbital can have the same quantum numbers O None of these statements is correct O In principle one cannot simultaneously determine the mass of a sub-atomic particle and its charge with equal accuracy. In other words, in contradistinction to the determinism of Classical physics, where in theory, given initial conditions, we should be able to predict all future behavior of all particles, quantum mechanics reveals an inherent indeterminacy built into nature and our understanding of it. O In principle one cannot simultaneously determine the position of a sub-atomic particle and its speed (or momentum) with equal accuracy. In other words, in contradistinction to the determinism of Classical physics, where in theory, given initial conditions, we should be able to predict all…

In his classic 1940 book Mr. Tompkins in Wonderland, physicist George Gamow imagined a trip to a "quantum jungle" where the value of Planck's constant h was 1.0 J*s instead of its real value of 6.63 x 10-34 J*s. Imagine that while exploring in this quantum jungle, you disturb a community of bats residing in a ruined temple. Imagine that a "beam" of identical bats (each with a mass of 0.5 kilograms) flies at 6 meters per second through two temple doors 3 meters apart and into a flat, large courtyard beyond. If you are 30 meters from the doors, where could you stand in the courtyard to avoid being struck by any bats? (Hint: the answer is 1.6 meters, 4.9 meters, etc, to either side of the center line perpendicular to the doors)