For example, in one dimension if we had two particles in a one-dimensional infinite potential well described by the following two wave functions.

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1 Identical particles In classical physics one can label particles in such a way as to leave the dynamics unaltered or follow the trajectory of the particles say by making a movie with a fast camera. Thus two configurations that are related by exchanging (identical) particles are not physically equivalent. In quantum mechanics it is impossible in principle to keep track of the trajectory without affecting it. Therefore, identical (same mass, charge, and other quantum numbers) particles are indistinguishable in nature; to wit, two physical realizations in which two identical particles have been switched are indistinguishable. This is true of electrons, protons, neutrons, photons, helium atoms (same isotope) etc. A careful statement was given by Messiah and Greenberg : The principle of indistinguishability is stated as states that differ only by a permutation of identical particles cannot be distinguished by any observation whatsoever. Let us examine what constraint this places on the wave function by examining a simple problem. Consider the wave function of two identical particles that can be specified by the coordinates only: Ψ( r, r ). We know that Ψ( r, r ) d 3 r d 3 r yields the probability of finding one particle in an infinitesimal volume d 3 r around r and another in an infinitesimal volume d 3 r around r. Now consider the wave function Ψ( r, r ). The probability density of this wave function should be the same as that for Ψ( r, r ) since the particles are identical and we still find one particle around r and the other around r. For example, in one dimension if we had two particles in a one-dimensional infinite potential well described by the following two wave functions Ψ (x, x ) = ( ) ( ) a sin πx πx sin and Ψ (x, x ) = ( ) ( ) a sin πx πx sin both of them should describe the same physical situation if the two particles are indistinguishable. The probability densities are clearly not the same. See Figure. We have also plotted the probability densities corresponding to the symmetric and antisymmetric combinations [ ( ) ( ) ( ) ( )] πx πx πx πx sin sin ± sin sin /. Let us denote by the position coordinate, define a permutation operator P that permutes the position coordinate, spin projection along the z-axis, etc of the particle A. Messiah and O. W. Greenberg, Phys. Rev., 36, B48-67, (964).

2 .0 x x Figure : For the simple product wave function the probability density is clearly not symmetric around the x = x line. The density at (0., ) is not the same as (, ) which it should be if the particles are identical..0 x x Figure : The symmetric wave function has a probability density that is symmetric around the x = x line. Observe that the two particles (bosons) are more likely to be at the same position.

3 .0 x x Figure 3: The antisymmetric wave function has a probability density that is symmetric around the x = x line. Observe that the two particles (fermions) avoid each other with the density vanishing along x = x. labeled. So we will use ψ(, ) as a shorthand for ψ( r, s ; r, s ). Define the permutation operator P that exchanges the position coordinate, spin along the z-axis, etc of two identical particles. Thus, P ψ(, ) = ψ(, ). This is a mathematical definition. An admittedly unsatisfactory but often used argument follows: If the two wave functions are equivalent then we demand that ψ(, ) = α ψ(, ) where α can be a complex number with unit modulus. Now consider P P ψ(, ) = P ψ(, ) = P (αψ(, )) = α ψ(, ) and thus α = ±. The two signs correspond to bosons (+, symmetric) and fermions(- antisymmetric) respectively. Electrons are spin-/ particles. Make sure you remember that s = /, means the wave function is an eigenstate of S with eigenvalue ( ) + h = 3 h /4 for all electrons. All electrons have the same mass, charge, lepton number, etc. The wave function contains information about the coordinate and the z-component of the 3

4 intrinsic angular momentum. If the electron is confined to a one-dimensional potential its wave function may be ( ) a sin(3πx/a) 0 i.e., it has spin up and is in the n = 3 state. It is convenient to write this φ 3 (x) +. One can obviously place the electron in a normalized superposition of such states. Next consider two non-interacting electrons and let us confine ourselves to one dimension for simplicity. (This is strictly not correct since there is no spin-statistics theorem in one dimension but this is a convenient assumption to illustrate the principles in a simple setting.) We will discuss three-dimensional wave functions soon. Consider two states that are eigenfunctions of a spin-independent Hamiltonian (i.e., H does not depend on spin variables), such as the harmonic oscillator or the infinite potential well. Label the states by nd b. We will first study wave functions Ψ(x, s ; x, s ) that can be written as a product ψ(x, x ) χ(s, s ) where χ(s, s ) represents the spin part of the wave function. Choose the spatial part of wave function of the two electrons to be φ a (x ) φ a (x ). If we did not have spin this state would not be allowed. Why not? We demand that Ψ(x, s ; x, s ) = Ψ(x, s ; x, s ) i.e., the wave function is antisymmetric under the exchange of both spin and space values. Our choice makes the spatial part symmetric. So for the total wave function to be allowed we make the spin part antisymmetric. The spin part of the two electron wave function has four possibilities s s : + +, +, +, and. Clearly, the first and the last kets are symmetric under the exchange of the spin values and will not do. The middle two do not have any particular symmetry under exchange since + +. However we can easily construct combinations ( ) which is symmetric under the exchange and ( + + ) which is antisymmetric. Thus an allowed wave function is φ a (x )φ a (x ) ( + + ). make sure you understand what two-spin kets go with the spatial part (φ a (x )φ b (x ) φ b (x )φ a (x )). 4

5 We will show next that the three symmetric kets (referred to as a triplet ) correspond to a spin state and the antisymmetric ket (called a singlet) corresponds to a spin 0 state. 5