QMPT 540
Bosons• Symmetric N-boson states • Symmetrizer
• Symmetrized states
• with • Completeness (unrestricted)
• Ordered sp quantum numbers • Normalization
• Not ordered
S =1
N !
�
p
P
|�1�2...�N ⇥ =�
N !n�!n�� !...
⇥1/2
S |�1�2...�N )�
�
n� = N�
�1�2...�N
n�!n�� !...N !
|�1�2...�N ⇥ ��1�2...�N | = 1
ordered�
�1�2...�N
|�1�2...�N ⇥ ��1�2...�N | = 1
��1�2...�N |��1�
�2...�
�N ⇥ = ��1|��
1⇥��2|��2⇥...��N |��
N ⇥= ⇥�1,��
1⇥�2,��
2...⇥�N ,��
N
��1�2..�N |��1�
�2..�
�N ⇥ =
1[n�!..n�� !..]1/2
�
P
��1|��p1
⇥��2|��p2
⇥..��N |��pN
⇥
QMPT 540
Boson addition and removal operator• Notation
• As for fermions
• Two particles when • Otherwise
• General case
• Contrast with fermions: for bosons commutation relations
• from the requirement to generate symmetric states
|�1�2...�N ⇤ =�
N !n�!n�� !...
⇥1/2
S |�1�2...�N ) � |n�n�� ...⇤
|�� = a†� |0�
|�⇥� = a†�a†⇥ |0� � �= ⇥
|��� = |n� = 2� =1⇤2a†�a†� |0�
|n�n⇥ ...n⇤� =1
[n�!n⇥ !...n⇤!]1/2
�a†�
⇥n�⇤a†⇥
⌅n⇥
...�a†⇤
⇥n⇤ |0�
[a�, a†⇥ ] = a�a†⇥ � a†⇥a� = ��,⇥
[a�, a⇥ ] = [a†�, a†⇥ ] = 0
QMPT 540
Properties of boson operators• As for oscillators
• and
• As for fermions
• and
a†� |n�n⇥ ...n⇤� =⇤
n� + 1 |n� + 1 n⇥ ...n⇤�
a� |n�n⇥ ...n⇤⇥ =⌅
n� |n� � 1 n⇥ ...n⇤⇥
F =�
�⇥
��| F |⇥⇥ a†�a⇥
V =12
�
�⇥⇤⌅
(�⇥|V |⇤⌅)a†�a†⇥a⌅a⇤
QMPT 540
Statistical mechanics• Large systems • Finite temperature
• Grand canonical ensemble
• Statistical operator • with
• and chemical potential
• Grand partition function
�G =e��(H�µN)
ZG
� = (kBT )�1
µ
ZG = Tr�e��(H�µN)
⇥
=⇤
N
⇤
n
��Nn | e��(H�µN) |�N
n ⇥ =⇤
N
⇤
n
e��(ENn �µN)
QMPT 540
Thermodynamic potential• Standard result • and therefore
• Ensemble averages
• Noninteracting systems replace • Summing over complete sets of states in Fock space can also be
accomplished by summing over all possible occupations of sp states in occupation number representation!
• Replace relevant operators by eigenvalues:
• Apply
�(T, V, µ) = �kBT lnZG
�G = e�(��H+µN)
�O⇥ = Tr��GO
⇥=
Tr�e��(H�µN)O
⇥
Tr�e��(H�µN)
⇥
H � H0
H0 |n1...n�� =�
i
ni�i |n1...n��
N |n1...n�� =�
i
ni |n1...n��Z0 =
�
n1...n�
⌅n1...n⇥| e��(H0�µN) |n1...n⇥⇧
=�
n1...n�
⌅n1...n⇥| e��(P
i ⇤ini�µP
i ni) |n1...n⇥⇧
=�
n1...n�
exp {�(µn1 � ⇤1n1)}... exp {�(µn⇥ � ⇤⇥n⇥)}
=⇥⇥
i=1
Tr (exp {��(⇤i � µ)ni})
QMPT 540
Noninteracting grand partition function• Yields
• with Tr including a summation over possible occupation numbers • Bosons: all occupation numbers possible
• Thermodynamic potential
• Average particle number
Z0 =�
n1...n�
exp {�(µn1 � ⇤1n1)}... exp {�(µn� � ⇤�n�)}
=�⇥
i=1
Tr (exp {��(⇤i � µ)ni})
ZB0 =
⇥⇥
i=1
⇥�
n=0
[exp {�(µ� ⇤i)}]n =⇥⇥
i=1
[1� exp {�(µ� ⇤i)}]�1
�B0 (T, V, µ) = �kBT ln
⇥⇥
i=1
[1� exp {�(µ� ⇤i)}]�1
= kBT⇥�
i=1
ln [1� exp {�(µ� ⇤i)}]
N = ⇥N⇤ = ��
⇥�B0
⇥µ
⇥
TV
⇧N⌃ ⇥��
i=1
n0i =
��
i=1
1exp {�(⇤i � µ)}� 1
QMPT 540
Noninteracting fermions at finite T• Restriction to 0 and 1 for occupation numbers
• Thermodynamic potential
• Particle number
ZF0 =
�⇥
i=1
1�
n=0
[exp {�(µ� ⇤i)}]n =�⇥
i=1
[1 + exp {�(µ� ⇤i)}]
�F0 (T, V, µ) = �kBT
��
i=1
ln [1 + exp {�(µ� ⇤i)}]
⇧N⌃ ⇥��
i=1
n0i =
��
i=1
1exp {�(⇤i � µ)} + 1
n0i =
1
exp {�("i � µ)}+ 1
QMPT 540
BEC in infinite systems• Ground state of noninteracting bosons: all in lowest sp level • This limit is approached when T → 0
• boson spectrum
• As before
• Transform to energy integral
• Thermodynamic potential
�(k) =�2k2
2m�
i
� �V
(2⇥)3
⇥dk
�V
(2⇥)34⇥k2 dk =
�V
2⇥2
�2m
�2
⇥3/2 ⇤d⇤
2⇤1/2=
�V
4⇥2
�2m
�2
⇥3/2
⇤1/2d⇤
�B0 = kBT
⇤V
4⌅2
�2m
�2
⇥3/2 ⇤ �
0d⇧ ⇧1/2 ln [1� exp {�(µ� ⇧)}]
= � ⇤V
4⌅2
�2m
�2
⇥3/2 23
⇤ �
0d⇧
⇧3/2
exp {�(⇧� µ)}� 1
QMPT 540
BEC• Energy
• Particle number
• Note so one confirms ideal gas
• Denominator represents occupation so may not become negative so chemical potential such that so here
• Fix density and lower temperature: should decrease
• The limit for
E =⇤
i
n0i ⇧i =
⇤V
4⌅2
�2m
�2
⇥3/2 ⌅ �
0d⇧
⇧3/2
exp {�(⇧� µ)}� 1
N =⇤
i
n0i =
⇤V
4⌅2
�2m
�2
⇥3/2 ⌅ �
0d⇧
⇧1/2
exp {�(⇧� µ)}� 1
� = �PV PV =23E
⇥� µ ⇥ 0 µ � 0|µ|
µ = 0 N =⇥V
4⇤2
�2m
�2
⇥3/2 ⇤ �
0d⌅
⌅1/2
exp {⌅/kBT0}� 1
=⇥V
4⇤2
�2mkBT0
�2
⇥3/2 ⇤ �
0dx
x1/2
exp (x)� 1
=⇥V
4⇤2
�2mkBT0
�2
⇥3/2
�(32)12⌅
⇤
QMPT 540
BEC• Rewrite
• with Riemann -function
• For temperatures below
integral only yields particles with
so those particles represented by • Remaining particles must all have
• Macroscopic occupation (~ ) of single state → BEC
T0 =3.31�2/3
�2
mkB
�N
V
⇥2/3
�(32) = 2.612 �
T0
� > 0
N�>0(T ) = N
�T
T0
⇥3/2
� = 0
N�=0(T ) = N
⇤1�
�T
T0
⇥3/2⌅
N
QMPT 540
BEC for 4He• Check that at there is a discontinuity in the slope of the
specific heat (see Fetter and Walecka) • For 4He with
• Experimental transition at and ...
• called λ transition • N0 ~ 10%
• Superfluidity ≠ideal gas but transition still related to BEC!?
T0
� = 0.145 g cm�3 T0 = 3.14 KT0 = 2.2 K
Tilley & Tilley Superfluidity & Superconductivity
QMPT 540
BEC in traps• Laser cooling & magneto-optical
trapping techniques • Evaporative cooling
• Atomic gases are metastable
• Why? • Temperatures
• Densities
• Scales 10s to 100s of
• Magnetic traps for alkali atoms look like harmonic oscillators with different oscillator lengths
500 nK to 2µKfew �1014atoms/cm3
µm
Vext(r) =12
m��2
xx2 + �2yy2 + �2
zz2⇥
QMPT 540
Oscillators• Eigenvalues • Ground state: all atoms in state with
• Wave function
• with • bosons in this sp state
• Calculate density grows with
• shape does not and is determined by trap potential: • Actual scale
• Finite temperature: atoms occupy excited states
• For use classical Boltzmann distribution
• spherical (Landau&Lifshitz) • If width
⇥nxnynz = (nx + 12 ) ��x + (ny + 1
2 ) ��y + (nz + 12 ) ��z
nx = ny = nz = 0
⇥000(r) =�m⇤HO
��
⇥3/4exp
⇤�m
2� (⇤xx2 + ⇤yy2 + ⇤zz2)
⌅
�HO = (�x�y�z)1/3
|�N0 � =
1⇤N !
(a†000)N |0�
�(r) = N |⇥000(r)|2N
N
aHO =�
�m�HO
⇥1/2
aHO � 1 µm
kBT � ��HO
�cl(r) ⇥ exp��Vext(r)
kBT
⇥
Vext(r) =12m�2
HOr2 RT = aHO
�kBT
��HO
⇥1/2
QMPT 540
BEC observation• BEC observed in the form of sharp peak in the center • Wave function in momentum space also Gaussian: width
• So both in coordinate space and momentum space
• Infinite system all particles zero momentum but no signature in coordinate space
• Observe velocity distribution/ density distribution
Anderson et al.
Science 269, 198 (1995)
Rubidium atoms (velocity)
left: just above condensation
middle: just after
right: further cooled
asymmetric trap
� a�1HO
QMPT 540
Trapped bosons at finite temperature• Interaction between bosons important for the actual shapes • Still useful considerations for noninteracting bosons
• Number of particles
• Energy
• Usual thermodynamic limit not possible • Still separate lowest state from the sum with
• As for infinite system: can be of order when
• with
• This limit is reached for a critical temperature
N =⇤
nxnynz
1exp
��(⇤nxnynz � µ)
⇥� 1
E =⇤
nxnynz
⇤nxnynz
exp��(⇤nxnynz � µ)
⇥� 1
N0
N µ� µc =32
�⇥
� = (�x + �y + �z)/3
T = Tc
QMPT 540
Convert to integrals• Rewrite particle number
• do numerically or for
• semiclassical description: excitation energies level spacing
• valid for large and
• Integrate: • Imposing at
• yields
• or
• Evaluate energy similarly etc.
• Interaction important as are finite size corrections
N �N0 =�
nx �=0,ny �=0,nz �=0
1exp {��(⇥xnx + ⇥yny + ⇥znz)}� 1
N �⇥N �N0 =
� �
0dnxdnydnz
1exp {��(⇥xnx + ⇥yny + ⇥znz)}� 1
�N kBT � ��HO
N �N0 = �(3)�
kBT
�⇥HO
⇥3
�(3) � 1.202
N0 � 0 Tc
kBTc = �⇥HO
�N
�(3)
⇥1/3
= 0.94 �⇥HO N1/3
N0
N= 1�
�T
Tc
⇥3