Positronium consists of an electron and a positron (both...

Name: Problem 7, Chapter 8: The Physics of Quantum Mechanics
Uploaded: 2020-08-10T16:18:13-08:00
Duration: 3 min 31 s
Channel: Tara Appleyard
Description: Positronium consists of an electron and a positron (both spin-half and of equal mass) in orbit around one another. What are its energy levels? By what factor is a positronium atom bigger than a hydrogen atom?

Positronium consists of an electron and a positron (both spin-half and of equal mass) in orbit around one another. What are its energy levels? By what factor is a positronium atom bigger than a hydrogen atom?

Key Concepts

Bohr Radius

The Bohr radius represents the typical scale of the orbital radius in a hydrogen-like atom and is determined by a formula that involves the reduced mass of the system. For positronium, with a reduced mass that is half that of hydrogen’s electron mass, the Bohr radius doubles compared to hydrogen’s Bohr radius. This change in scale explains why the positronium atom is larger than a hydrogen atom, providing insight into how fundamental constants and particle mass influence atomic dimensions.

Reduced Mass

In a two-body quantum system like positronium, both particles contribute equally to the dynamics. The reduced mass (?) is used to simplify the analysis by converting the two-body system into an equivalent one-body problem. For systems where both masses are equal, such as the electron-positron pair in positronium, the reduced mass becomes half of the individual mass. This adjustment is crucial for accurately determining energy levels and characteristic radii in quantum models.

Bohr Model Energy Levels

The Bohr model, extended to hydrogen-like atoms, provides a formula for the energy levels that is inversely proportional to the square of the principal quantum number (n) and directly dependent on the reduced mass of the system. In positronium, because the reduced mass is half that of the hydrogen atom, the energy levels are scaled accordingly. This yields energy eigenvalues that are half the magnitude (in absolute value) of the corresponding levels in the hydrogen atom, illustrating the impact of mass distribution on quantum energies.

Transcript

00:01 In this question, we are looking at calculating the ground state energy of a positronium atom, or the ground state ionization energy of a positronium atom.

00:13 So i've begun by writing out boz energy equation here on the board.

00:18 And by looking at it, we can see that most of this equation is constants, like the e or epsilon naught, but some parts aren't.

00:30 So the bits that we need to know are the z, the charge of the nucleus, the mass, and the n.

00:37 So the charge of the nucleus in our case, we're going to say is 1, because we are treating the positron as if it is our nucleus.

00:48 So z is equal to 1, so we can rewrite the top of the equation as minus m .e to the 4.

00:54 And then on the bottom of the equation, we know that n is equal to 1 because we're in the ground state.

01:00 So we have 8 h -bar squared, epsilon -not squared.

01:06 Then the only thing that we need to work out in this equation that is not a constant is the mass.

01:12 And we need the mass of the combined system.

01:16 So we need to use the reduced mass, which is the effective inertial mass of our two particles rotating together.

01:24 So reduced mass mu is found by doing the first mass times by the second, divided by the first plus the second.

01:33 Now in our problem, we have a positron and an electron, and they have the same mass.

01:38 So we can rewrite this as m e squared over 2me...

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