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2.2: System of Identical Bosons

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  • ln this section we construct a set of basis vectors for a system of \(n\) identical bosons. The boson system is treated analogously to the treatment of the fermion system in Section 2.1.

    The subscript n on I r/(l) > has be added to serve as a ¡emi¡de¡ that the vector is in l'!f,. and 22

    Basis Vectors for the one-Boson system


    \[ | \beta_r \rangle_a\]

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

    \[ \sum_{r=1}^{\infty} | \beta_r \rangle \langle \beta _r | = 1_{\alpha}\]

    \[ _{\alpha} \langle \beta_r | \beta_{s} \rangle_{\alpha} = \delta_{rs}\]


    Example of basis vectors

    The | þ, > may, for example, be chosen to be the simulta¡eous eigenvectors of the Hamiltonian for a thee-dimensional non¡elativistic harr¡onic oscillato¡ t -¡2 --' /-:\2 -- and of (J ) and J3 where the eigenvatue of l,í)- is s(s * l)fi.2 whe¡e -" is \ / \-/ --' an integer. .s is the intrinsic spin of the elementa4r boson.l Tbe I p, > may also be si.EultaEeous eigenve.ctors of tbe iDþ¡Þat variables cha¡.ge, baryon oumber, leptoE Dumber, isospi¡, strangetress a¡d cba¡m. .As iD tbe ferEio! case i¡ Section 3.1, we Deed Eot specify tbese va¡iables he¡e. lþ,ì (3. r 7) I I d" t. tl, l:ln < í1, | þ, >- 6," ad (3.18) (3.19) 1a 2.

    General one-boson state

    The general one-boson state at time \(t\) is

    \[ | \psi(t) \rangle_{\alpha} = \sum_{r=1}^{\infty} \psi_r(t) | \beta_r \rangle_{\alpha} \label{3.20} \]


    \[,þ,(t) :< p, l rþ(t) > tld \label{3.21}\]

    is the probability amplitude rhar the boson is in the søte lp" > at time \t\).

    Basis vectors for the n-boson system

    The n-particle Hilbert space *i is a tenso¡ product (2.22) of n identicai sp:rces. When s is an integer, it is spanned by vectors of the form

    l {t, ìl p" >r ... 1& > (3.22)

    where particle I is in single-particle state I B, >, pattrcle 2 is in single-particle state I p" > and particle n is in single-particle staæ | p¿ >. The r¿-boson Hilbert space b*i Q.28) is spanned by symmetric combinations of vectors of the form (3.22).'Ihat is, ö'Fi is spanned by vectors of the form lþ,ì lp">2 ; t&7 vyhe¡e r'(.s(...(f (3.24) and where sum det denotes a determinant which has plus signs in its definition rather than minus signs.



    We use a slight difference in notation to denote n-fermion and n-boson basis vectors: The lefÌ side of (3.7) has {...} a¡d the left side of (3.23) has [...]. This notation anúcipates anticommutâto¡s fo¡ a fermion system in Chapters 4 and 5 and commutators for a boson system in Chapær 6.

    2. Manifest symmetry (3.23) is manifestly symmetric under panicle interchange. 25

    3. Occupied- ¡lþlqs The set of single-particle labels r,s,...,t tells which single-particle st¿tes are occupied. Because of the symmetrizing, one cannot specify which particle occupies which state.

    4. No Pauli Exclusion Principle

    There is no Pauli Exclusion Principle for a system of identical bosons because ,sur¡ det does not vanish if all elements of one ¡ow are equal to all elements of another row, that is, if two o¡ mo¡e of r', s, . . . , f are equal,

    5. Occupation numbers Let n, be the number of particles occupying the single-particle state I B" >. Then o!n,!n (3.2s) ¿¡d Ð", = '¡1. (3.26) rr, is the occupation number for the sirgle-particle state I B, >. (3.23) is labelled by the occupation numbers rLltTL2t.. . In contradistinction to the fermion case, all bosons may occupy one single-particle state. 6. Basis for the n-boson Hilbert space The set of vectors (3.23) is orthononnal and spans ò'1,",. That is, b | ¡ n¡r,r, ...)>< nln¡,2...1 l= I, ?¿l t¡2... (3.27) bnù Ð: DÐ o*,, 1L)rl2 . t¿l =0 t¿2:0 (3.28) ,r' -= 5- rr, j=1 (3.2e) Fu¡thermore, (3.30) The representation provided by the set of vecto¡s (3.23) is called the occupation numbe¡ represgnt¿tion for the z-boson system. 7. General ¡¿-boson state The general r¿-boson st¿te has the form ò I c'{?) >: )- I ,fu,,,,...l >< 1) - ' 't¡ t¡. 117t12.. nlr4n2...Jl,l'(Ð ¡ (3.31) where < nln1n2 ' . .l I 'i (¿) ì (3.32) is the probability amplitude at time t that rz1 bosons occupy the single-particle state | /31 ), and rr2 bosons occupy the single-particle state i É2 >, and so on