35. Special Relativity

A particle is described by the wave function where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.

Physicists first attempted to understand the hydrogen atom by applying the laws of classical physics. Consider an electron of mass m and charge −e in a circular orbit of radius r around a proton of charge +e. The minimum energy needed to ionize a hydrogen atom (i.e., to remove the electron) is found experimentally to be 13.6 eV. From this information, what are the electron's speed and the radius of its orbit?

A classical atom that has an electron orbiting at frequency ⨍ would emit electromagnetic waves of frequency ⨍ because the electron's orbit, seen edge-on, looks like an oscillating electric dipole. What is the total mechanical energy of this atom?

What is the speed of a particle when its kinetic energy equals its rest energy? Does the mass of the particle affect the result?

A photoelectric-effect experiment finds a stopping potential of 1.56 V when light of 200 nm is used to illuminate the cathode. From what metal is the cathode made?

A 1.5-μm-wavelength laser pulse is transmitted through a 2.0-GHz-bandwidth optical fiber. How many oscillations are in the shortest-duration laser pulse that can travel through the fiber?

A stick of length ℓ₀, at rest in reference frame S, makes an angle θ with the x axis. In reference frame S', which moves to the right with velocity $\overrightarrow{v}$ = vî with respect to S, determine (a) the length l of the stick, and (b) the angle θ it makes with the x' axis.

Consider a single-slit diffraction experiment using electrons. (Single-slit diffraction was described in Section 33.4.) Using Figure 39.5 as a model, draw A graph of |ψ(x)|² for the electrons on the detection screen.

Figure 37.7 identified the wavelengths of four lines in the Balmer series of hydrogen. Predict the wavelength of the fifth line in the spectrum.

A 16-nm-long box has a thin partition that divides the box into a 4-nm-long section and a 12-nm-long section. An electron confined in the shorter section is in the n = 2 state. The partition is briefly withdrawn, then reinserted, leaving the electron in the longer section of the box. What is the electron’s quantum state after the partition is back in place?

Calculate the de Broglie wavelength of a $5.00$-g bullet that is moving at $340$ m/s. Will the bullet exhibit wavelike properties?

1.0×10⁶ atoms are excited to an upper energy level at t = 0 s. At the end of 20 ns, 90% of these atoms have undergone a quantum jump to the ground state. What is the lifetime of the excited state?

The graph in FIGURE P38.42 was measured in a photoelectric-effect experiment. What is the work function (in eV) of the cathode?

FIGURE P39.31 shows the wave function of a particle confined between x = 0 nm and x = 1.0 nm. The wave function is zero outside this region. Draw a graph of the probability density P(x)=|ψ(x)|²

(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box of width $0.360$ nm.

(b) The electron makes a transition from the $n=1$ to $n=4$ level by absorbing a photon. Calculate the wave­length of this photon.