A positronium atom consists of a positron and an electron. In Bohr-like model, the two particles

A positronium atom consists of a positron and an electron....

Key Concepts

Spectral Transitions and Emission Wavelengths

The emission or absorption of light in atoms results from electrons transitioning between energy levels, with the wavelength of the emitted photon determined by the energy difference between these levels. In systems like positronium, modifications in the energy levels due to changes in reduced mass lead to shifts in the corresponding transition wavelengths compared to hydrogen. Understanding these transitions is essential in spectroscopy and in identifying the spectral signatures of different atomic systems.

Positronium

Positronium is an exotic atom-like bound state consisting of an electron and its antiparticle, the positron. It is analogous to the hydrogen atom but with the proton replaced by a positron. This system is useful for studying quantum electrodynamics (QED) effects and for understanding two-body problems where both constituents have equal mass.

Reduced Mass

Reduced mass is a concept used to simplify two-body problems by converting them into an equivalent one-body problem. It is defined as the product of the masses divided by their sum. In systems like positronium, where the two particles have equal mass, the reduced mass becomes half the mass of either particle, adjusting the dynamics of orbital motion compared to single-particle systems like hydrogen.

Bohr Model

The Bohr model is a semi-classical description of the atom that quantizes the angular momentum of the electron. It proposes that electrons move in fixed circular orbits, with their radii and energies determined by quantization conditions. Although it was historically applied to hydrogen, its principles can be extended to similar two-body systems such as positronium by incorporating the reduced mass in the calculations.

Orbital Radius in the Ground State

In the Bohr model, the orbital radius of an electron in the ground state is determined by a balance between the centripetal force and the Coulomb attraction, and it scales inversely with the reduced mass. For systems like positronium, using the reduced mass instead of the electron mass from the hydrogen atom leads to a modification of the ground-state orbital radius compared to hydrogen.

Ground-State Energy

The ground-state energy in the Bohr model is derived from the quantization of orbits and is dependent on the reduced mass of the system. It represents the binding energy of the electron in the atom and is negative, indicating that energy must be provided to remove the electron from its bound state. For positronium, replacing the electron mass with its reduced mass in the energy formula leads to a different value than that of hydrogen.

Transcript

0:00 Hi there.

00:01 So for this problem we have a positron atom that consists of a positron in an electron.

00:07 And the board -like model, the two particles rotate in circle about their common center of mass.

00:15 So for this, the first part of this problem, we need to calculate the reduced mass of a positronian atom in terms of the mass of an electron.

00:25 So we need to find a, the mass of the positron, so we are going to say that is the total mass.

00:36 Now, the equation or the formula for the reduced mass is given like the following.

00:45 Because the system is composed of an electron in a proton, we are going to call the mass of the proton mp and the mass of the electron m .e.

00:55 So the formula for the reduced mass is mp times m e, the mass of the proton times the mass of the electron over the sum of these masses, the mass of the proton plus the mass of the electron.

01:13 Now, we know that the proton and the electron, the positron, sorry, the positron is an antiparticle of the electron, is an antiparticle of the so the only thing that changes is its charge in comparison to the electron so and their masses are the same so the masses of the positron is the same as the mass of the electron so with this we can simplify this border so we can say that we are going to tame in the numerator that the mass of the electron squared and in the denominator we will have two times the mass of the electron so we can simplify this order by eliminating one of the one in the numerator or one in the denominator.

02:06 So we will have that this is equal.

02:08 The mass, the reduced mass is the mass of the electron half of the mass of an electron.

02:14 So that's the reduced mass of this system known as positronian.

02:22 Now for part b of this problem, we are told to determine the orbital radius of a it's ground state electron.

02:32 So for this we use the following equation to obtain radius r1 because we want that when the ground state, so the ground state it's n equal to one.

02:49 We know the equation for rn, for any given value of the state.

02:57 So we will have that this is n squared times the plan constant squared over b times the total or the reduced mass times the charge of the electron.

03:19 So as you can see, this reduced for r1 as the following.

03:24 When we introduce one in here, we'll obtain one in there, so we'll have this.

03:29 Sorry, oh, this is not bar h, it's just bar.

03:34 It's just h.

03:36 So this corresponds, and this at the same time, is equal to two times the atomic radius.

03:48 So we know the value for the atomic radius.

03:52 That is...

03:57 This is a sub -series also known as the bor's radio, and has a value of 5 .29.

04:09 Sub series 5 .29 times 10 to the minus 11 meters.

04:24 So when we plot that in the equation for r1, we obtain that this is two times that value.

04:31 So we obtain that the radius for the ground stain n equals to 1.

04:39 It's approximately 0 .106 nanometers.

04:47 So that's a solution for par b of this problem.

04:51 Now for par c of this problem, we are asked to find its ground state energy...

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