Ch 41: Atomic Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 41: Atomic Physics

Problema 31

Capitolo 41, Problema 31

For an electron in the 1s state of hydrogen, what is the probability of being in a spherical shell of thickness 0.010a_B at distance (a) ½ a_B, (b) a_B, and (c) 2a_B from the proton?

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Understand the problem: The probability of finding an electron in a spherical shell is determined by the radial probability density function, which is derived from the square of the radial wavefunction multiplied by the volume element. For the hydrogen atom in the 1s state, the radial wavefunction is given by \( R_{1s}(r) = \frac{2}{a_B^{3/2}} e^{-r/a_B} \), where \( a_B \) is the Bohr radius.

Write the expression for the radial probability density: The radial probability density is \( P(r) = |R_{1s}(r)|^2 \cdot 4\pi r^2 \), where \( |R_{1s}(r)|^2 \) is the square of the radial wavefunction. Substituting \( R_{1s}(r) \), we get \( P(r) = \frac{4}{a_B^3} e^{-2r/a_B} \cdot 4\pi r^2 \).

Set up the probability for a spherical shell: The probability of finding the electron in a thin spherical shell of thickness \( dr \) at a distance \( r \) is \( dP = P(r) \cdot dr \). Substituting \( P(r) \), we have \( dP = \frac{16\pi}{a_B^3} r^2 e^{-2r/a_B} dr \).

Evaluate the probability for each case: For each distance \( r \) (\( r = \frac{1}{2}a_B, a_B, 2a_B \)), substitute the value of \( r \) into the expression \( dP = \frac{16\pi}{a_B^3} r^2 e^{-2r/a_B} dr \) and use \( dr = 0.010a_B \) to calculate the probability for the corresponding spherical shell.

Perform the calculations: For each case, simplify the expression by substituting \( r \) and \( dr \) into \( dP \). This will yield the probability for the electron to be in the specified spherical shell at the given distance. Note that the final numerical evaluation can be done separately if needed.

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Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It introduces concepts such as wave-particle duality and quantization of energy levels, which are essential for understanding the behavior of electrons in atoms, including their probability distributions.

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Hydrogen Atom Model

The hydrogen atom model describes the structure of the hydrogen atom, where an electron orbits a proton. In this model, the electron's position is described by wave functions, which provide the probability density of finding the electron at various distances from the nucleus, particularly in the 1s state, which is the lowest energy level.

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Probability Density

Probability density in quantum mechanics refers to the likelihood of finding a particle, such as an electron, in a specific region of space. For the hydrogen atom, the probability density is derived from the square of the wave function's amplitude, and it is crucial for calculating the probability of the electron being found within a given spherical shell at various distances from the nucleus.

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For an electron in the 1s state of hydrogen, what is the probability of being in a spherical shell of thickness 0.010aB at distance (a) ½ aB, (b) aB, and (c) 2aB from the proton?

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For an electron in the 1s state of hydrogen, what is the probability of being in a spherical shell of thickness 0.010a_B at distance (a) ½ a_B, (b) a_B, and (c) 2a_B from the proton?