An electron has a de Broglie wavelength of $2.80\times {10}^{-10}$ m. Determine (a) the magnitude of its momentum and (b) its kinetic energy (in joules and in electron volts).

Through what potential difference must electrons be accelerated if they are to have:

(a) the same wavelength as an x ray of wavelength $0.220$ nm; and

(b) the same energy as the x ray in part (a)?

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

An electron is moving with a speed of $8.00\times {10}^6$ m/s. What is the speed of a proton that has the same de Broglie wavelength as this electron?

An alpha particle ($m=6.64\times {10}^{-27}$ kg) emitted in the radioactive decay of uranium-$238$ has an energy of $4.20$ MeV. What is its de Broglie wavelength?

For crystal diffraction experiments (discussed in Section $39.1$), wavelengths on the order of $0.20$ nm are often appropriate. Find the energy in electron volts for a particle with this wavelength if the particle is a photon.

(a) An electron moves with a speed of $4.70\times {10}^6$ m/s. What is its de Broglie wavelength?

(b) A proton moves with the same speed. Determine its de Broglie wavelength.

(a) The $x$-coordinate of an electron is measured with an uncertainty of $0.30$ mm. What is the x-component of the electron's velocity, $v_x$, if the minimum percent uncertainty in a simultaneous measurement of $v_x$ is $1.0\%$?

(b) Repeat part (a) for a proton.

A scientist has devised a new method of isolating individual particles. He claims that this method enables him to detect simultaneously the position of a particle along an axis with a standard deviation of $0.12$ nm and its momentum component along this axis with a standard deviation of $3.0\times {10}^{-25}$ kg-m/s. Use the Heisenberg uncertainty principle to evaluate the validity of this claim.