# Correct understanding of symmetrization requirement?

• dracobook
In summary, when discussing how the wave function for a fermion needs to be antisymmetric, Griffiths seems to derive the Pauli exclusion principle by showing that the wave function can be antisymmetric when the particle states are the same, but the total wave function needs to be antisymmetric. However, on page 210 he seems to pull out of thin air the fact that the whole wave function has to be antisymmetric for fermions and symmetric for bosons.
dracobook
Hi all,

I want to make sure I have the right understanding of the symmetrization requirement...in particular what is discussed in section 5.1 in Griffiths' Introduction to Quantum Mechanics 2nd edition. When we have a two-electron state, the wave function (described by the product both its position wave function and spinor) has to be antisymmetric with exchange of the electron. Since the singlet spinor function is antisymmetric, it must be combined with a symmetric spatial function in order for it to properly describe a fermion (in this case an electron).

I am just really confused because Griffiths seems to derive that just the position wave function is antisymmetric for fermions and symmetric for bosons on pages 203-205.

However, on page 210 he seems to pull out of thin air the fact that the whole wave function has to be antisymmetric for fermions and symmetric for bosons and the position wave function for a fermion can be BOTH symmetric or antisymmetric depending on the corresponding spinor.

Any help with the clarification would be greatly appreciated.
Thanks,
D

of course,it should be the total wave function for fermion to be antisymmetric when pauli principle is taken into account.

yes. But Griffiths derives Pauli exclusion principle by showing that when the positional wave functions are antisymmetric with exchange, then the wave function cancels when the particle states are the same.

you will have to multiply the positional wave function by those spin wave function to obtain the total wave function.

For fermions the N-body Hilbert space of indistinguishable particles is constructed by the totally antisymmetrized tensor products of N single-particle states. Let's discuss non-relativistic quantum mechanics, where spin and momentum (or spin and position) are compatible observables.

An electron has spin 1/2, and a nice single-particle basis is given by the position-spin basis, $|\vec{x},\sigma \rangle$, where $\vec{x} \in \mathbb{R}^3$ and $\sigma \in \{-1/2,+1/2 \}$.

Now let's look at the Hilbert space for two electrons. It consists of all superpositions of antisymmetrized single-particle states, i.e.,
$$|\Psi \rangle=\frac{1}{\sqrt{2}} (|\psi_1 \rangle \otimes |\psi_2 \rangle - |\psi_2 \rangle \otimes |\psi_1 \rangle).$$
The wave two-particle wave function is given by
$$\Psi(\vec{x}_1,\sigma_1;\vec{x}_2,\sigma_2) = \frac{1}{\sqrt{2}} \left [\psi_1(\vec{x}_1,\sigma_1) \psi_2(\vec{x}_2,\sigma_2)-\psi_2(\vec{x}_1,\sigma_1) \psi_1(\vec{x}_2,\sigma_2) \right].$$
Here
$$\psi_j(\vec{x},\sigma)=\langle \vec{x},\sigma|\psi_j \rangle$$
are the single-electron wave functions.

Of course you can also use a basis with good total spin $S$. For two spin-1/2 particles the total spin can be either 0 or 1. There is only one state with total spin 0:
$$|\vec{x}_1,\vec{x}_2,S=0,\Sigma=0 \rangle = \frac{1}{\sqrt{2}} (|\vec{x}_1, 1/2 \rangle \otimes |\vec{x}_2,-1/2 \rangle - | \vec{x}_1,-1/2 \rangle \otimes |\vec{x}_2,-1/2 \rangle.$$
The $S=0$ part of the above antisymmetrized product state is thus
$$\Psi(\vec{x}_1,\vec{x}_2,S=0,\Sigma=0)=\frac{1}{2} \left [\psi_1(\vec{x}_1,1/2) \psi_2(\vec{x}_2,-1/2)-\psi_2(\vec{x}_1,1/2) \psi_1(\vec{x}_2,-1/2) - \psi_1(\vec{x}_1,-1/2) \psi_2(\vec{x}_2,1/2)+\psi_2(\vec{x}_1,-1/2) \psi_1(\vec{x}_2,1/2) \right].$$
It's obviously antisymmetric, when interchanging the single-particle spins and symmetric, when interchanging the single-particle positions. Alltogether it's still totally antisymmetric.

The same you can do for the $S=1$ basis,
$$|\vec{x}_1,\vec{x}_2,S=1,\Sigma=1 \rangle=|\vec{x}_1,1/2 \rangle \otimes \vec{x}_2,1/2 \rangle,$$
$$|\vec{x}_1,\vec{x}_2,S=1,\Sigma=0 \rangle = \frac{1}{\sqrt{2}} \left (|\vec{x}_1,1/2 \rangle \otimes |\vec{x}_2,-1/2 \rangle + |\vec{x}_1,-1/2 \rangle \otimes |\vec{x}_2,1/2 \rangle \right ),$$
$$|\vec{x}_1,\vec{x}_2,S=1,\Sigma=-1 \rangle=|\vec{x}_1,-1/2 \rangle \otimes \vec{x}_2,-1/2 \rangle.$$
The two-particle wave function for all three states is of course symmetric under exchange of the single-particle spins and thus necessarily antisymmetric under exchange of the single-particle positions. In total it's antisymmetric under single-particle-state exchange.

Any other two-body state is given as a superposition of several such antisymmetrized two-particle product states.

dracobook, this excerpt from Townsend's excellent QM book should help clarify things.

## 1. What is the symmetrization requirement in science?

The symmetrization requirement in science refers to the principle that physical laws and theories should remain unchanged under certain transformations, such as rotations or reflections. This ensures that the laws are consistent and valid regardless of the position or orientation of an object.

## 2. Why is the symmetrization requirement important in scientific research?

The symmetrization requirement is important because it allows scientists to make accurate predictions and draw reliable conclusions about the behavior of physical systems. Without this requirement, inconsistencies and contradictions may arise, leading to flawed theories and inaccurate results.

## 3. How is the symmetrization requirement applied in different fields of science?

The symmetrization requirement is a fundamental principle that is applied in various fields of science, such as physics, chemistry, and biology. In physics, it is used to describe the behavior of particles and the laws of motion. In chemistry, it is used to understand the symmetry of molecules and their properties. In biology, it is used to study the symmetry of organisms and their structures.

## 4. Can the symmetrization requirement be violated?

In some cases, the symmetrization requirement may be violated, such as in the presence of external forces or due to the limitations of certain measurement tools. However, scientists strive to minimize these violations and ensure that the underlying principles of symmetry are maintained in their research and theories.

## 5. How does the symmetrization requirement relate to the laws of physics?

The symmetrization requirement is closely tied to the laws of physics, as it is a fundamental principle that governs the behavior of physical systems. The laws of physics, such as the laws of conservation, are derived from the symmetries of the universe and are essential for understanding the fundamental nature of our world.

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