Obtain the relativistic correction ∝p^4 to the nonrelativistic kinetic energy of an electron, an

Obtain the relativistic correction ∝p^4 to the...

Key Concepts

Nonrelativistic Kinetic Energy Expansion

In quantum mechanics, the kinetic energy of a particle is commonly approximated by the nonrelativistic expression p²/(2m), which is valid when the particle’s velocity is much less than the speed of light. However, when relativistic effects become significant, the exact expression for kinetic energy can be expanded in powers of p²/(mc)². The first correction term in this expansion is proportional to p?, accounting for relativistic deviations from the classical kinetic energy at low velocities.

Relativistic Correction to Kinetic Energy

The relativistic correction term, proportional to p?, emerges as the next order term when expanding the relativistic energy expression in a power series. This term, often written as -p?/(8m³c²), corrects the nonrelativistic Hamiltonian. It is crucial for calculations where small relativistic effects contribute to the energy shifts of quantum systems, such as in the hydrogen atom.

First-Order Perturbation Theory

First-order perturbation theory is a method in quantum mechanics used to estimate the effect of a small additional term (perturbation) in the Hamiltonian on the energy levels of a system. By treating the relativistic correction as a perturbation, one can calculate the shift in energy levels by taking the expectation value of the perturbing term with respect to the unperturbed wavefunctions.

Expectation Value in the Hydrogen Atom Ground State

In the hydrogen atom, the ground state wavefunction is well-known, which facilitates the evaluation of expectation values of operators such as p?. By computing the expectation value of the relativistic correction term using the hydrogen ground state wavefunction, one obtains the energy shift in the ground state due to relativistic effects. This approach combines the methods of perturbation theory with the specific properties of the hydrogen atom's wavefunctions to quantify small corrections to its energy spectrum.

Transcript

00:01 Hi there.

00:02 So for this problem, we are told that the relativistic shift in the energy levels of a hydrogen atom due to the relativistic dependence of mass on velocity can be determined by using the atomic agent functions to calculate the expectation value for the relative energy.

00:22 Of the quantity where the relative energy is equal to the relative energy minus the classical and the relativistic energy minus the classical.

00:30 Energy.

00:31 So the difference between the relativistic and classical expressions for the total energy.

00:37 So we need to show that for the momentum not too large, we obtain that is this expression in here for the relativistic energy.

00:48 So with that said, we start by defining that the relativistic energy as the sum of the kinetic energy and the potential energy.

00:58 So we know that the, the, the kinetic energy, it's relativistic.

01:05 And then we're gonna have the following form.

01:08 That is the mass times the speed of light square, which is the minus the res energy, which is its res mass times the speed of light.

01:19 Now we know that the mass increase with the speed, so we need to write the mass m in terms of the laurence factor.

01:30 So we're gonna have that this is m, 0 times the speed of light and this times 1 divided by the square root of 1 minus beta square.

01:42 We take the square root of this and this minus 1.

01:48 Now with this we know that beta is defined in this case as the speed divided by the speed of light.

01:57 So the relativistic momentum p the relativistic momentum is defined as the mass times the speed.

02:05 So we know that the mass in this case, again, we can write it as m -0 times the speed of lie times beta and beta times one minus beta square and this elevated to minus one divided by two.

02:25 So to express the kinetic energy in terms of the momentum, we know that the momentum prime is equal to the momentum divided by m sub zero times the speed of light.

02:37 So we will have that this is beta times one minus beta square and this elevated to minus one divided by two.

02:47 So we know that we can write the lorentz factor then solving from this.

02:54 We can write that one minus beta square elevated to minus one divided by two is equal to 1 plus the momentum prime to the square.

03:09 So that we will have that the kinetic energy is equal to m sub 0 times the speed of light and this times 1 plus the momentum prime to the square and this elevated to 1 over 2 minus 1.

03:28 So we can write this, we can expand this, and so that will be m -suk 0 times the speed of light.

03:36 And this times 1 plus 1 divided by 2 times p prime to the square, minus 1 divided by a times p prime to the 4, and so on for the other terms in here.

03:53 So with that said, and the last term in here that we're going to have is minus 1 after all the other terms, the infinite terms in here...

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