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2.1: System of identical fermions and Slater Determinants

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  • In this section we construct a set of basis vectors for a system of n identical fermions.

    Basis vectors for the one-fermion system

    \[ | \phi_r \rangle_{\alpha} \label{3.1} \]

    where \(r =1, 2, \ldots ,\infty\).be a complete orthonormal set of vectors spanning the Hilbert space '¡.l for fermion \(\alpha\). That is

    \[ \sum_{r=1}^{\infty} | \phi_r \rangle \langle \phi _r | = 1_{\alpha} \label{3.2}\]

    \[ _{\alpha} \langle \phi_r | \phi_{s} \rangle_{\alpha} = \delta_{rs} \label{3.3}\]

    where \(1_{\alpha}\) is the unit operator in the one-particle space for fermion number \(\alpha\)


    L Notation: subscript c¡ The subscript a on | /, > and 1o has been added to serve as a rcminder that d the vectors and operators a¡e i¡ the Hilbert space t{.,rf for fennion number a. 2. Denumerable set of basis vectors It is convenient to use a denume¡able set of vecto¡s to span the one-fermion Hilbert space. We use coordinate/spin kets I im" >, momentum/spir kets | 1nz, > and momentum/helicity kets I h^ (Ð > in Chapær 5 to span the one-fermion HiÌbert space. These kets, which are also a¡e discussed ín QLB: Some Inrentz InvarianÍ Systems, a¡e each labelled by a continuous variable. 3. Example of basis vectors The | ç;, > may, for example, be chosen to be the simultaneous eigenvectors of the Hamiltonian for a th¡ee-dimensional non¡elativistic harmonic oscillatot z -l? and of f ./l and Jr whe¡e J: X x P *,9 whe¡e X, P and 5 are the \./ Cartesian position, momentum and spin of the particle.l / =,\2 The eigenvalue of | 5 | is s(s _ì- f)ñ.' where .s ls a hall-odd rnteger. .s ls the " \./ intrinsic spin of the elementary fer¡nion. 4. General one-fermion state The general one-fermion stâte at time i is ) Tbe I d. ) may also be simultaDeous eigebveators of tbe i¡æmal variables charge, baryon numbe¡, leptoD DuEber, isospi-n, strangeness and cha¡m, l e ¡eed uot sPeciJy tbese vaÍiables here. t7 I 'i (r) ì= 1,,p,{t) I ó, r=l (3.4) and ,þ,(t) :< ö, I tl'(t) > dat (3.s) is the probability ampütude thar the fermion is in the sr¿te I d, > at time ¿. Basis vectors for the n-fermion system The z-particle Hilbert space {..ri is a tensor producr (2.22) of n identical spaces. When .s is a half-odd integer, it is spanned by vectors of the form I l,-'>l ,A >...l,Á- -' I Yr -.1t Y5 -2 t ro -r, l? Á\ where particle I is in single-particle state I d" >, pa¡ticle 2 is in single-panicle state | ç4s > and panicte n is in single-panicle staæ | /¿ >. The ¡¿-fermion Hilbert space f Aí, Q.zg) is spanned by antisymmetric combinations of vectors of the fon¡ (3.6). That is, /*i is spanned by vectors of the form l8 (3.7) ló,>t¿ ld" > t¿ lóu> t¿ |,, \,,1,,2 ) 7: {*)* .", I i i' i "ï i ltø'; lö,> where ?'. Then r¿r:0or1 (3.e) mC Ë,":, (3.10) ir, is the occupation numbe¡ for the single-particle staæ | /" >. 20 (3.7) is labelled by the occupation numbe¡s ,t:'i )n2,.... ?. Basis for the n-fermion Hilbert space The set of vectors (3.7) is a¡ orthononnal bæis fo¡ f¡,". That isJ r | ¡rr1rr,n,'..) >< rtltt1n2. "i l= t, Í11112". (3.1r) lll \- - \- \-... ¡ ¿._ _ z--- z_ v111r, tlrÍt2... t¿t =0 t¿2=0 (3.12) whe¡e ,,' = \-rr, ZrJ (3. r 3) Furthermore, < n{n1n2-..} | nþt'1n!2 .} ;: 6,,,,,\6,,,,,, ... (3.14) The subscript ?¿ oD I, bas be added to serve as a ¡emi¡der thal the unil operaûor is i.D tÌ¡; 21 The representation provided by the set ofvectors (3.7) is called the occupation number representation fo¡ the n-fermion system. 8. General n-fermion state The general r¿-fermion state has the form2 f I ?,1'(r) >: | ¡ n1rr1rr2....} >< n{n4'r2...} I '/(f) > (3.15) ' ¡, /----r I 1LtrL2.. < n{n1n2...} | 'l(l) ì (3. i 6) is the probability amplìtude at time t that rz1 fermions occupy the singleparticle state I dl >, and z2 fermions occupy the single-particle stÃtl | ö2 >, and so on.