According to quantum mechanics, the electron cloud for a...

According to quantum mechanics, the electron cloud for a hydrogen atom in the ground state has a charge density $$\rho(r)=\frac{q}{\pi a^{3}} e^{-2 r / a},$$ where $q$ is the charge of the electron and $a$ is the Bohr radius. Find the atomic polarizability of such an atom. [Hint: First calculate the electric field of the electron cloud, $E_{e}(r) ;$ then expand the exponential, assuming $r \ll a$. For a more sophisticated approach, see W. A. Bowers, $A m$. J. Phys. 54, 347 (1986).]

Key Concepts

Atomic Polarizability

Atomic polarizability is a measure of how easily the electron cloud of an atom can be distorted by an external electric field. This results in an induced dipole moment and influences the atom’s interaction with electromagnetic fields. It is central to understanding intermolecular forces, van der Waals interactions, and various optical properties of matter.

Electron Charge Density

In quantum mechanics, the electron cloud is described by a charge density function that gives the probability distribution of the electron's location around the nucleus. For the hydrogen atom in its ground state, the specific exponential form of the charge density encapsulates how the electron is spatially distributed, and it dictates how the electric field is generated by the electron cloud.

Bohr Radius

The Bohr radius is a fundamental physical constant that sets the characteristic length scale for the hydrogen atom in quantum mechanics. It appears in the expressions for the electron charge density and governs the spatial extent of the electron cloud, thereby playing a crucial role in calculations regarding atomic structure and interactions.

Electric Field and Series Expansion

Calculating the electric field due to the electron cloud involves integrating the contribution from the charge density. In regions close to the nucleus (r much less than the Bohr radius), the exponential part of the charge density can be expanded in a Taylor series. This expansion simplifies the mathematical treatment and is essential for approximating the field and consequently determining the induced dipole moment and polarizability.

Transcript

00:01 So polarizability is a constant that measures how easy it is to turn the molecules or atoms inside of a substance into dipoles when they are placed in an external electric field.

00:21 So p in this relationship is the dipole moment, electric diapy.

00:30 And basically, here we'll draw a little picture.

00:34 And molecules are by and large neutral, but if they are placed in an external electric field, there is a force that pushes the positive charge inside the atom or molecule to one side, and the negative charge goes to the other side.

00:57 And so there forms a dipole moment p, which is proportion, proportional to q, q being the plus minus charge on either location, times d, and we'll put a little vector sign on that, because the d is in a certain direction along the line of the external electric field.

01:28 So our goal here is to understand how this happens inside of a hydrogen atom.

01:35 So we are going to start with the density of the electron cloud as being a ball, a spherical ball.

01:50 So we're looking basically at the 1 -s state.

01:56 And in this, the a is the bore radius, which gives a scale for how large the hydrogen atom is.

02:10 So we're assuming the hydrogen is not excited.

02:15 And the idea is, okay, so we'll draw the cloud, and supposedly the positive charge is at the center of that cloud.

02:31 And what supposedly happens when the electric field comes in is we are going to keep the cloud neutral, not neutral, but spherical.

02:43 Yes, thank you.

02:45 But we are going to displace the positive charge a certain amount from the center of that cloud.

02:54 We'll call that d, and the electron charge still has a center in the middle of that sphere.

03:08 So there's a displacement by that d that we're most interested in.

03:14 And the idea is that the displacement happens until the external field acting on our positive proton is balanced by the electric field of the electron cloud that is trying to pull it back towards the center.

03:41 So that is a simplifying assumption is that those two balance.

04:02 Okay, so our goal will be to find the electron electric field due to this distribution of charge and use it in our polarizability.

04:19 So basically we are going to be setting qd equal to alpha times the electron.

04:28 Electric field and we are going to solve for the alpha.

04:35 The way that we will find the electric field of the electron is we will use gauss's law and we're going to make some pretty crude approximations, which probably will get us off a little bit.

04:53 Okay, but here we are inside the electron cloud and assuming that it still stays spherical and we're going to draw little gaussian surface out to where the proton is.

05:10 And of course, it's that gaussian surface that we are using to calculate our electric field, and we'll call it distance d out from the center.

05:21 So remember what gauss's law says.

05:24 It says that the electric flux through that closed surface is proportional to the enclosed charge.

05:41 Okay.

05:42 And we're assuming a spherical electric field.

05:45 So on the left -hand side, we are not going to do that integral.

05:49 We're going to simply set that flux equal to the radial electric field strength times the surface area of that green surface.

06:01 So 4 pi d squared.

06:05 And let's see about the enclosed charge on the other side.

06:11 So we want to take our density and multiply it over that spherical volume of that gaussian surface.

06:25 So that's where the green surface comes in.

06:29 And we will have to use spherical coordinates along with the density...

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