Question

Consider the first-order correction to the energy of interacting spins illustrated in Example Problem 28.3 for \(\psi_{2}\) Calculate the energy correction to the wave functions \(\psi_{1}=\alpha(1) \alpha(2), \psi_{2}=\beta(1) \alpha(2),\) and \(\psi_{4}=\beta(1) \beta(2)\) Show that your results are consistent with \(\Delta E=m_{1} m_{2} h J_{12}\) with \(m_{1}\) and \(m_{2}=+1 / 2\) for \(\alpha\) and \(-1 / 2\) for \(\beta\).

Short answer

We have calculated the energy corrections for the given wave functions using perturbation theory and found that the results are consistent with the given formula \(\Delta E = m_1 m_2 h J_{12}\) for interacting spins. We obtained the energy corrections as follows: \[\Delta E_1^{(1)} = \frac{1}{4} h J_{12}, \quad \Delta E_2^{(1)} = -\frac{1}{4} h J_{12}, \quad \text{and} \quad \Delta E_4^{(1)} = \frac{1}{4} h J_{12}.\]

Step-by-step solution

Perturbation Theory and Hamiltonian

The Hamiltonian for the interacting spins can be written as: \[H = H_0 + H'\] where the unperturbed Hamiltonian: \[H_0 = h (J_1 + J_2)\] and the perturbation Hamiltonian: \[H' = h J_{12}\] We will use perturbation theory to find the energy correction for each state. The first-order correction in energy is given by: \[\Delta E_n^{(1)} = \langle \psi_n | H' | \psi_n \rangle\] Now let's calculate the energy correction for each of the given wave functions.

Energy Correction for \(\psi_1\)

Let's find the energy correction for \(\psi_1 = \alpha(1) \alpha(2)\): \[\Delta E_1^{(1)} = \langle \psi_1 | H' | \psi_1 \rangle = \langle \alpha(1) \alpha(2) | hJ_{12} | \alpha(1) \alpha(2) \rangle\] Now, using the formula: \[\Delta E = m_{1} m_{2} h J_{12}\] Substitute \(m_1 = m_2 = +1/2\): \[\Delta E_1^{(1)} = \frac{1}{2} \cdot \frac{1}{2} h J_{12} = \frac{1}{4} h J_{12}\]

Energy Correction for \(\psi_2\)

Let's find the energy correction for \(\psi_2 = \beta(1) \alpha(2)\): \[\Delta E_2^{(1)} = \langle \psi_2 | H' | \psi_2 \rangle = \langle \beta(1) \alpha(2) | hJ_{12} | \beta(1) \alpha(2) \rangle\] Now, using the formula: \[\Delta E = m_{1} m_{2} h J_{12}\] Substitute \(m_1 = -1/2\) and \(m_2 = 1/2\): \[\Delta E_2^{(1)} = \left(-\frac{1}{2}\right) \cdot \frac{1}{2} h J_{12} = -\frac{1}{4} h J_{12}\]

Energy Correction for \(\psi_4\)

Finally, let's find the energy correction for \(\psi_4 = \beta(1) \beta(2)\): \[\Delta E_4^{(1)} = \langle \psi_4 | H' | \psi_4 \rangle = \langle \beta(1) \beta(2) | hJ_{12} | \beta(1) \beta(2) \rangle\] Now, using the formula: \[\Delta E = m_{1} m_{2} h J_{12}\] Substitute \(m_1 = m_2 = -1/2\): \[\Delta E_4^{(1)} = \left(-\frac{1}{2}\right) \cdot \left(-\frac{1}{2}\right) h J_{12} = \frac{1}{4} h J_{12}\]

Conclusion

We have calculated the energy corrections for the given wave functions as: \[\Delta E_1^{(1)} = \frac{1}{4} h J_{12}\] \[\Delta E_2^{(1)} = -\frac{1}{4} h J_{12}\] \[\Delta E_4^{(1)} = \frac{1}{4} h J_{12}\] This is consistent with the given formula \(\Delta E = m_1 m_2 h J_{12}\) for the interacting spins.

Key concepts

Interacting Spins

In quantum mechanics, spins are a fundamental property of particles, similar to charge or mass. When spins interact, it typically involves a system where two or more spins influence each other's energy states. This interaction can be described using a Hamiltonian, a mathematical operator that quantifies the energy of a system.

For interacting spins, we often consider scenarios where there are specific couplings between spins. In our exercise, spins from two particles interact, and their interaction is captured through terms involving their product, such as \( m_1 m_2 h J_{12} \), where \( m_1 \) and \( m_2 \) are magnetic quantum numbers for the two spins, and \( J_{12} \) denotes the interaction strength.

This means that the energy levels of these spins will depend on their relative orientations due to the interaction term. Whether these spins align or oppose affects the energy corrections to their quantum states.

Energy Correction

When discussing energy corrections in quantum systems, we focus on how additional factors influence the energy levels that would exist in a simple, unperturbed system.

In perturbation theory, the first-order energy correction is used to approximate how a small disturbance or "+perturbation," typically denoted by \( H' \), shifts the energy of an eigenstate \( \psi_n \) of the unperturbed Hamiltonian \( H_0 \).

The formula for this first-order energy correction is \( \Delta E_n^{(1)} = \langle \psi_n | H' | \psi_n \rangle \). This means we calculate the overlap, or expectation value, of the perturbation with itself in the given state.

In the specific context of interacting spins, the energy correction depends on the quantum numbers \( m_1 \) and \( m_2 \) of the spin states, such that \( \Delta E = m_1 m_2 h J_{12} \). This captures how the interaction term \( h J_{12} \) changes the energy based on the product of spins.

First-Order Correction

First-order correction is an essential tool in perturbation theory, which analyzes the effects of a small perturbation on a system's energy levels and states.

The theory assumes that the perturbation is small enough not to drastically alter the system's fundamental nature but sufficient to influence quantifiable properties like energy. In our exercise, we use the first-order correction for an approximate solution to the energy levels modified by the interaction term.

Mathematically, first-order corrections simplify to finding expectation values in the basis of the unperturbed eigenstates. For example, for wave functions like \( \psi_1 = \alpha(1) \alpha(2) \) and \( \psi_2 = \beta(1) \alpha(2) \), the first-order corrections are given by \( \langle \alpha(1) \alpha(2) | H' | \alpha(1) \alpha(2) \rangle \) and \( \langle \beta(1) \alpha(2) | H' | \beta(1) \alpha(2) \rangle \), respectively.

The corrections help understand how wave functions and energies shift due to interactions defined by \( H' \). This is crucial in predicting the behavior of quantum systems under external influences.

Wave Functions

Wave functions are central to quantum mechanics, representing the probability amplitude of finding a particle in a set of positions and times.

For interacting spins, wave functions encompass the spin states of particles involved, such as \( \psi_1 = \alpha(1) \alpha(2) \), depicting both particles in "up" spin states. Each combination reflects a unique quantum state, and computing overlaps with perturbation operators helps determine energy corrections.

The notations \( \alpha \) and \( \beta \) refer to the standard states of spin-1/2 particles, "up" and "down" respectively. Each wave function like \( \psi_1 \), \( \psi_2 \), and \( \psi_4 \) in the original exercise represent different alignments or configurations of spins.

Mathematically, these states are treated as vectors in a complex space, and they interact with external operators (like \( H' \)) to yield physical quantities such as energy. The study of wave functions in such exercises enables us to predict the possible outcomes when quantum systems experience interactions.