The probability that the hydrogen atom electron is inside its nucleus.

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Chapter 39, Problem 66P

To determine

To find:

The probability that the hydrogen atom electron is inside its nucleus.

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In atoms there is a finite, though very small, probability that, at some instant, an orbital electron will actually be found inside the nucleus. In fact, some unstable nuclei use this occasional appearance of the electron to decay by electron capture. Assuming that the proton itself is a sphere of radius 1.1 * 10-15 m and that the wave function of the hydrogen atom’s electron holds all the way to the proton’s center, use the ground-state wave function to calculate the probability that the hydrogen atom’s electron is inside its nucleus.

(a) An electron and a 0.0400 kg bullet each have a velocity of magnitude 510 m/s, accurate to within 0.0100%. Within what lower limit could we determine the position of each object along the direction of the velocity? (Give the lower limit for the electron in mm and that for the bullet in m.) for the electron 0.01136 for the bullet 2.585e-34 mm m (b) What If? Within what lower limit could we determine the position of each object along the direction of the velocity if the electron and the bullet were both relativistic, traveling at 0.450c measured with the same accuracy? (Give the lower limit for the electron in nm and that for the bullet in m.) for the electron for the bullet 4.2899 X nm 9.76445e-42 X m

Consider a very simplistic model of atomic nucleus in 1D: a proton is completely localized in a 1D box of width L = 1.00 × 10¬14m. In other words, the proton wavefunction outside of the "nucleus" is zero. Note that L represents a typical nuclear radius. (A) What are the energies of the ground and the first excited states? If the proton makes a transition from the first excited state to the ground state, what is the angular frequency of the emitted photon? (B) What is the probability that the proton in its ground state (i.e., the lowest energy state) is not found in the distance L/12 around each boundary of the box? (C) Using the uncertainty principle, derive a minimum possible value on the momentum uncertainty in the second state above the ground state. (D) Compare your answer to the previous question (B) to probability distribution one would obtain for a classical particle. First argue about how the probability distribution would look for a classical object in its ground state. How…