Question

Exercise 44 gives an antisymmetric multiparticle state for two particles in a box with opposite spins. Another antisymmetric state with spins opposite and the same quantum numbers is $$ \psi_{n}\left(x_{1}\right) \downarrow \psi_{n}\left(x_{2}\right) \uparrow-\psi_{n}\left(x_{1}\right) \uparrow \psi_{n}\left(x_{2}\right) \downarrow $$ Refer to these states as \(\mid\) and 11 . We have tended to characterize exchange symmetry as to whether the state's sign changes when we swap particle labels. but we could achieve the same result by instead swapping the particles' stares, specifically the \(n\) and \(n^{\prime}\) in equation \((8-22)\). In this exercise. we look at swapping only parts of the state-spatial or spin. (a) What is the exchange symmeiry - symmetric (unchanged). antisymmetric (switching sign), or neither-of multiparticle states 1 and \(\mathrm{II}\) with respect to swapping spatial states alone? (b) Answer the same question. but with respect to swapping spin states/arrows alone. (c) Show that the algebraic sum of states 1 and \(\mathrm{II}\) may be written \(\left(\psi_{n}\left(x_{1}\right) \psi_{n}\left(x_{2}\right)-\psi_{n}\left(x_{1}\right) \psi_{n}\left(x_{2}\right)\right)(\downarrow T+\uparrow \downarrow)\) where the left arrow in any couple represents the spin of particle 1 and the right arrow that of particle 2 (d) Answer the same questions as in parts \((a)\) and (b). but for this algebraic sum. (e) Is the sum of states I and 11 still antis ymmetric if we swap the particles? total-spatial plus spin -states? (f) If the two particles repel each other, would any of the three multiparticle states - l. II, and the sum - be preferred? Explain.

Short answer

The exchange symmetry for spatial states switching is symmetric, while for the swapping spin states is antisymmetric. The algebraic sum of states follows the same pattern. Even when both spatial and spin states are swapped, the antisymmetry is preserved. If the particles repel each other, antisymmetric multi-particle states tend to be preferred.

Step-by-step solution

Identify the symmetry type when spatial states are swapped

When swapping the spatial states alone, one must interchange \(x_1\) and \(x_2\). In other words, in multiparticle states I and II, replace \(x_1\) by \(x_2\) and vice versa. Since both states are symmetric with respect to this operation, ('changing nothing'), the exchange symmetry is symmetric.

Identify the symmetry type when spin states are swapped

The interchange of spin states is equivalent to swapping up \(\uparrow\) and down \(\downarrow\) spin. For instance, in multiparticle state I, interchanging down \(\downarrow\) and up \(\uparrow\) spin has the effect of reversing the state such that the sign flips. Therefore, with respect to swapping of spin states alone, the multiparticle states I and II are antisymmetric.

Show the algebraic sum of states

The algebraic sum of states can be obtained by summing up state I and II. After factoring out common terms, the algebraic sum takes the form \[\left(\psi_{n}\left(x_{1}\right)\psi_{n}\left(x_{2}\right)-\psi_{n}\left(x_{1}\right)\psi_{n}\left(x_{2}\right)\right)(\downarrow \uparrow+\uparrow \downarrow) \]

Classify symmetry for algebraic sum (spatial and spin states)

The result in part (c) can be analyzed in the same manner as done in parts (a) and (b). When it comes to swapping spatial states or spin states, the algebraic sum of states I and II retains the same type of exchange symmetry, meaning that it is symmetric when swapping spatial states and antisymmetric when swapping spin states.

Verify antisymmetry for total state swapping

When the particles - including their spins and their spatial states - are switched, the overall state remains the same, and so the sum of states I and II is still antisymmetric.

Effect if the two particles repel each other

If two particles repel each other, their energies are higher when they are closest together. In such cases, antisymmetric states are generally preferred since they satisfy Pauli's exclusion principle that precludes identical particles being in the same state. It means that there is a natural tendency to keep them apart, thus reducing the magnitude of the repulsion.

Key concepts

Multiparticle States

When we dive into the fascinating world of quantum mechanics, we are introduced to the concept of multiparticle states. These refer to the description of a system containing two or more particles. The complexity arises because the particles are indistinguishable and can't be labeled as we do in the classical world. For example, when considering electrons, we talk about the overall state of the system rather than specifying each electron individually.

The exercise proposes a scenario where we have two electrons (particles) in a box, each with their own specific quantum state. We represent their combined state mathematically, which includes both their spatial position, represented by \(\psi_{n}(x)\), and their spin states, indicated by up \(\uparrow\) and down \(\downarrow\) arrows. The way in which these states combine is instrumental in determining the overall behavior of the electron pair.

Exchange Symmetry

The principle of exchange symmetry addresses the behavior of the multiparticle state when we switch the individual states of the particles—in our case, the electrons. If swapping the states of two particles doesn't change the overall state, then it exhibits symmetry. However, if swapping the states results in a sign change of the wave function, then the system manifests antisymmetry.

In the given exercise, we see that by interchanging the spatial states of the electrons (switching \(x_1\) and \(x_2\)), the composite wave function does not change, indicating symmetry with respect to spatial state exchange. On the other hand, exchanging the spin states (flipping \(\uparrow\) and \(\downarrow\)) results in a sign change of the wave function, revealing antisymmetry with respect to spin state exchange.

Antisymmetry

Delving deeper, antisymmetry is a crucial concept, especially when considering fermions, such as electrons. Antisymmetric states are such that when we exchange two fermions, the sign of the wave function flips, which completely alters the wave function's value. This arises from the fermionic nature that requires particles to obey the Pauli exclusion principle.

The provided solution illustrates this antisymmetry by showing that swapping the spin states of the two particles in each multiparticle state leads to a wave function with an opposite sign. This characteristic is fundamental in quantum mechanics and has profound implications, including the fact that no two identical fermions can occupy the same quantum state simultaneously—a statement that becomes clearer as we explore the Pauli exclusion principle.

Pauli Exclusion Principle

The Pauli exclusion principle is a quantum mechanics cornerstone applicable to all fermions, which states that no two identical fermions can occupy the same quantum state within a quantum system simultaneously. This principle explains the antisymmetry observed in the exercise. It forms the basis for understanding the electronic structure of atoms and the behavior of particles in everyday matter.

When considering the algebraic sum of states I and II in our exercise, we continue to see the effects of this principle. Whether we swap spatial or spin positions, the Pauli exclusion principle enforces the overall antisymmetry of the wave functions for the multiparticle states. This inherently averts the possibility that both electrons will find themselves in exactly the same state, providing a deeper understanding of the spatial distribution and energies of electrons in a given system.