In terms of the position vectors 𝐱α, 𝐱1 and 𝐱2 of the αparticle and two electrons, the centre of

step 1 In conclusion, we are given potential energy u equal to alpha x square plus beta y square and we

In terms of the position vectors 𝐱α, 𝐱1 and 𝐱2 of the...

In terms of the position vectors $\mathbf{x}_{\alpha}, \mathbf{x}_{1}$ and $\mathbf{x}_{2}$ of the $\alpha$ particle and two electrons, the centre of mass and relative coordinates of a helium atom are $$ \mathbf{X} \equiv \frac{m_{\alpha} \mathbf{x}_{\alpha}+m_{\mathrm{e}}\left(\mathbf{x}_{1}+\mathbf{x}_{2}\right)}{m_{t}}, \quad \mathbf{r}_{1} \equiv \mathbf{x}_{1}-\mathbf{X}, \quad \mathbf{r}_{2} \equiv \mathbf{x}_{2}-\mathbf{X} $$ where $m_{t} \equiv m_{\alpha}+2 m_{\mathrm{e}}$. Write the atom's potential energy operator in terms of the $\mathbf{r}_{i}$ Show that $$ \begin{gathered} \frac{\partial}{\partial \mathbf{X}}=\frac{\partial}{\partial \mathbf{x}_{\alpha}}+\frac{\partial}{\partial \mathbf{x}_{1}}+\frac{\partial}{\partial \mathbf{x}_{2}} \\ \frac{\partial}{\partial \mathbf{r}_{1}}=\frac{\partial}{\partial \mathbf{x}_{1}}-\frac{m_{\mathrm{e}}}{m_{\alpha}} \frac{\partial}{\partial \mathbf{x}_{\alpha}} \quad \frac{\partial}{\partial \mathbf{r}_{2}}=\frac{\partial}{\partial \mathbf{x}_{2}}-\frac{m_{\mathrm{e}}}{m_{\alpha}} \frac{\partial}{\partial \mathbf{x}_{\alpha}} \end{gathered} $$ and hence that the kinetic energy operator of the helium atom can be written $$ K=-\frac{\hbar^{2}}{2 m_{t}} \frac{\partial^{2}}{\partial \mathbf{X}^{2}}-\frac{\hbar^{2}}{2 \mu}\left(\frac{\partial^{2}}{\partial \mathbf{r}_{1}^{2}}+\frac{\partial^{2}}{\partial \mathbf{r}_{2}^{2}}\right)-\frac{\hbar^{2}}{2 m_{t}}\left(\frac{\partial}{\partial \mathbf{x}_{1}}-\frac{\partial}{\partial \mathbf{x}_{2}}\right)^{2} $$ where $\mu \equiv m_{\mathrm{e}}\left(1+2 m_{\mathrm{e}} / m_{\alpha}\right)$. What is the physical interpretation of the third term on the right? Explain why it is reasonable to neglect this term.

In terms of the position vectors $\mathbf{x}_{\alpha}, \mathbf{x}_{1}$ and $\mathbf{x}_{2}$ of the $\alpha$ particle and two electrons, the centre of mass and relative coordinates of a helium atom are
$$
\mathbf{X} \equiv \frac{m_{\alpha} \mathbf{x}_{\alpha}+m_{\mathrm{e}}\left(\mathbf{x}_{1}+\mathbf{x}_{2}\right)}{m_{t}}, \quad \mathbf{r}_{1} \equiv \mathbf{x}_{1}-\mathbf{X}, \quad \mathbf{r}_{2} \equiv \mathbf{x}_{2}-\mathbf{X}
$$
where $m_{t} \equiv m_{\alpha}+2 m_{\mathrm{e}}$. Write the atom's potential energy operator in terms of the $\mathbf{r}_{i}$
Show that
$$
\begin{gathered}
\frac{\partial}{\partial \mathbf{X}}=\frac{\partial}{\partial \mathbf{x}_{\alpha}}+\frac{\partial}{\partial \mathbf{x}_{1}}+\frac{\partial}{\partial \mathbf{x}_{2}} \\
\frac{\partial}{\partial \mathbf{r}_{1}}=\frac{\partial}{\partial \mathbf{x}_{1}}-\frac{m_{\mathrm{e}}}{m_{\alpha}} \frac{\partial}{\partial \mathbf{x}_{\alpha}} \quad \frac{\partial}{\partial \mathbf{r}_{2}}=\frac{\partial}{\partial \mathbf{x}_{2}}-\frac{m_{\mathrm{e}}}{m_{\alpha}} \frac{\partial}{\partial \mathbf{x}_{\alpha}}
\end{gathered}
$$
and hence that the kinetic energy operator of the helium atom can be written
$$
K=-\frac{\hbar^{2}}{2 m_{t}} \frac{\partial^{2}}{\partial \mathbf{X}^{2}}-\frac{\hbar^{2}}{2 \mu}\left(\frac{\partial^{2}}{\partial \mathbf{r}_{1}^{2}}+\frac{\partial^{2}}{\partial \mathbf{r}_{2}^{2}}\right)-\frac{\hbar^{2}}{2 m_{t}}\left(\frac{\partial}{\partial \mathbf{x}_{1}}-\frac{\partial}{\partial \mathbf{x}_{2}}\right)^{2}
$$
where $\mu \equiv m_{\mathrm{e}}\left(1+2 m_{\mathrm{e}} / m_{\alpha}\right)$. What is the physical interpretation of the third term on the right? Explain why it is reasonable to neglect this term.

Key Concepts

Interpretation of the Coupling (Cross-Derivative) Term

The third term in the kinetic energy operator, which involves the square of the difference between the derivatives with respect to the two electron coordinates, originates from the transformation and coupling of the electronic degrees of freedom relative to the centre of mass. Physically, this term represents a correlated kinetic energy component due to the nontrivial connection between the electrons’ motion, influenced by the presence of a finite-mass nucleus. It is effectively a measure of the residual coupling that remains after separating the centre of mass motion.

Differential Operators in Coordinate Transformations

The transformation from particle coordinates to centre of mass and relative coordinates affects the differential operators in the kinetic energy. Each derivative with respect to an original coordinate is expressed in terms of derivatives with respect to centre of mass and relative coordinates. This results in additional cross-terms that arise due to the dependencies among the coordinates. Understanding this transformation is key to correctly formulating the kinetic energy operator in the new coordinate system.

Approximation Justified by Mass Ratio

In atomic systems like helium, where the nucleus (alpha particle) is much heavier than the electrons, the impact of the coupling term is significantly reduced because it is scaled by the inverse of the total mass, which is dominated by the nucleus. Due to this large mass disparity, the contribution of the cross-derivative term to the total kinetic energy is very small. This justifies neglecting the term as a reasonable approximation, analogous to the assumptions made in the Born-Oppenheimer approximation, which separates the motion of heavy nuclei from that of the lighter electrons.

Centre of Mass and Relative Coordinates

In a system of multiple particles, including atoms, one often transforms the particle coordinates into a centre of mass coordinate and one or more relative (or internal) coordinates. The centre of mass coordinate represents the weighted average position of all the particles, while the relative coordinates describe the positions of the individual particles with respect to the centre of mass. This transformation is crucial for separating overall translation from internal motion, thereby simplifying the analysis of the system's dynamics.

Kinetic Energy Operator Separation

When rewriting the kinetic energy operator in terms of new coordinates, the operator separates into parts associated with the centre of mass motion and the internal (relative) motion. The process involves transforming the derivatives from the original particle coordinates to the new coordinate system. This separation not only facilitates the solution of the Schrödinger equation for complex systems but also highlights how different types of motion, such as collective translation and internal rearrangements, contribute to the overall kinetic energy.

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