For hydrogen atoms, the wave function for the state n=3, ℓ=0, mℓ=0 is ψ300=(1)/(81 √(3 π))((1)/(

For hydrogen atoms, the wave function for the state n=3,...

For hydrogen atoms, the wave function for the state $n=3$, $\ell=0, m_{\ell}=0$ is $$ \psi_{300}=\frac{1}{81 \sqrt{3 \pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2}\left(27-18 \sigma+2 \sigma^{2}\right) e^{-\sigma / 3} $$ where $\sigma=r / a_{0}$ and $a_{0}$ is the Bohr radius $\left(5.29 \times 10^{-11} \mathrm{~m}\right)$. Calculate the position of the nodes for this wave function.

For hydrogen atoms, the wave function for the state $n=3$, $\ell=0, m_{\ell}=0$ is
$$
\psi_{300}=\frac{1}{81 \sqrt{3 \pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2}\left(27-18 \sigma+2 \sigma^{2}\right) e^{-\sigma / 3}
$$
where $\sigma=r / a_{0}$ and $a_{0}$ is the Bohr radius $\left(5.29 \times 10^{-11} \mathrm{~m}\right)$. Calculate the position of the nodes for this wave function.

Key Concepts

Quantum Numbers

Quantum numbers are used in quantum mechanics to label the distinct states of a quantum system. For the hydrogen atom, the principal quantum number (n) determines the energy level, the orbital angular momentum quantum number (l) specifies the shape of the orbital, and the magnetic quantum number (m?) indicates the orientation of the orbital in space. Together, these numbers define the wavefunction and its properties, including the presence and number of nodes.

Radial Wavefunction

The radial wavefunction is the part of the overall wavefunction that depends on the distance from the nucleus. In the hydrogen atom, after separating variables in the Schrödinger equation, the full solution is constructed as a product of a radial part and an angular part. The radial function typically contains polynomial factors and an exponential decay factor, and its zeros correspond to specific radial distances, which are interpreted as nodes.

Nodes in Wavefunctions

Nodes are points or surfaces where the probability density of finding the electron is zero. In the context of the hydrogen atom, radial nodes refer to the radii at which the radial wavefunction equals zero. The number of radial nodes is given by n - l - 1. These nodes are important because they reveal the underlying quantum mechanical structure of the electron's behavior in the atom.

Bohr Radius

The Bohr radius is a fundamental physical constant that provides the scale for the size of the hydrogen atom in the Bohr model and quantum mechanical treatments. It appears in the expressions for the hydrogen atom wavefunctions and serves as the unit of distance when expressing the radial coordinate. Its presence in the wavefunction emphasizes the link between classical and quantum descriptions of atomic structure.

Transcript

00:01 In this problem, we have to find the position of specific nodes of our atom.

00:07 Now, we're given an equation specifically a wave function, and we have to use this to find the position of the nodes.

00:14 And remember, a node is where we would see zero electron density.

00:19 So let's review the equation we're given.

00:21 So this is a hydrogen atom with the principal quantum number 3, l is zero, and m subl is zero.

00:28 Remember, these are all quantum numbers.

00:30 So si of 300 or 300 would be equivalent to 1 over 81 times the square root of 3 pi, times 1 over a0 raised to the 3 halves.

00:42 We multiply that by 27 minus 18 sigma plus 2 sigma squared times e raised to negative sigma over 3.

00:51 Now sigma is equivalent to r times a not, and a0 is our bore radius.

00:57 So a not would be equivalent to 5 .29.

01:00 Times 10 to the negative 11th meters.

01:03 That's simply a constant.

01:04 That's just the bore radius for a hydrogen atom.

01:09 So now what we can do is we can replace everywhere we see sigma with r over a0.

01:16 And then remember, we're finding the nodes where this is zero electron density.

01:20 So we can set our entire equation equal to zero.

01:24 So we'd have zero equivalent to 1 over 81 times a square root of 3 pi times 1 over a0 raised to the 3 halves times 27 minus 18 times r over a0 plus 2 times r over a0 squared times e raised to negative r over 3 times a0.

01:45 Remember we just simply plugged in r over a0 wherever we saw a sigma...

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