高等固体物理-波色爱因斯坦凝聚

i
e
i / kT
i dpi
dpi
e
i / kT

b e
pi2 b pi2 b
pi2 dpi dpi
e
1 , kT
分母：


pi2 b

e
pi2 b
dpi

b
1

1
分子：


e
dp i [ b
E ( q1 ,..., q f ; p1 ,..., p f ) / kT
称为正则分布
对多自由度体系，有按 自由度的能量分配定理 ： 自由度i， i ~ p i2 : i bp i2 E i E ' (q1 ,..., q f ; p1 ,..., p i 1 , p i 1 ,... p f ) E '是除自由度i能量以外的能量，它与 i 无关： p

The Material For BEC

BEC was found in alkali metals e.g. 87Rb (铷), 23Na (钠), 7Li (锂) because: They are bosons. Each atom is a small magnetic compass, so that a cooling technique called magnetic cooling can work. The atoms have a small repulsion, so that they do not liquefy or solidify down to a very low temperature.
c R1B 2 (1 n2 ) R1 2 B c R21 (1 n1 ) R21 B R 2 1 n1 1 n1 e E1 / kT B n2 1 n2 e E 2 / kT R1 2
n1 n2 e E1 / kT e E2 / kT e (T ) ;只与温度有关的常数 1 n1 n2 1 n 1 n e
如粒子动力学只容许分立能量：
对量子谐振子： nh E
E
nhe
n 0

nh / kT
e
n 0

nh / kT
令x h / kT ,
分母： e x e 2 x ... 1
1 1 ex
d nh / kT h e x 分子： －h ( e ) dx n0 (1 e x )2
精细平衡：
ni Ri j
nj R ji
ni R i j n j R j i
对所有能级成立。 经典统计，有：
n1 e ( E1 E2 ) / kT n2
c n1 R21 e E1 / kT c E2 / kT n2 R12 e
对玻色子，有增强因子1+n:

]

1
b 2 3 / 2
i
1 1 kT 2 2
2 i
与b无关！
1 如 i cq , 则 i kT，与c无关。 2
一个服从经典统计的系统处于温度T的热平衡中，其能量 的每一独立的平方项都有平均值1/2kT.
考虑由电子和质子组成 的等离子体，虽然它们 质量不同 相当于b不同), ( 但平均动能相等。 不同硬度的大量谐振子 体系(相当于c不同), 其平均势能也相同。
专题三：玻色－爱因斯坦凝聚
1. 经典统计分布函数 2. 量子统计分布函数 3. 玻色－爱因斯坦凝聚
1. 经典统计分布函数
一绝热系统处在势场中：
x x dx y y dy z z dz p z p z dp z
p x p x dp x
p y p y dp y
E
h e
h / kT
1
普朗克能量均分定理
利用它可推导出黑体辐射公式
2. 量子统计分布函数
两个全同粒子， ,
B (1,2) F (1,2) 1 2 1 2 [ (1) ( 2) ( 2) (1)] [ (1) ( 2) ( 2) (1)] 2 (1) (1)
h mv
Against Our Intuition?!
In most everyday matter, the de Broglie wavelength is much shorter than the distance separating the atoms. In this case, the wave nature of atoms cannot be noticed, and they behave as particles. The wave nature of atoms become noticeable when the de Broglie wavelength is roughly the same as the atomic distance. This happens when the temperature is low enough, so that they have low velocities. In this case, the wave nature of atoms will be described by quantum physics, e.g. they can only stay at discrete energy states (energy quantization).
De Broglie 德布罗意 (1929 Nobel Prize winner) proposed that all matห้องสมุดไป่ตู้er is composed of waves. Their wavelengths are given by

= de Broglie wavelength h = Planck’s constant 普朗克常数 m = mass v = velocity
E / kT
e (T )
nB ( E )
1 e e E / kT 1
玻色－爱因斯坦分布函数


0
n B ( E ) N ( E )dE N
总粒子数
是和粒子数密度和温度 有关的量，
且和系统能级密度有关 。
费米子：1－n取代1+n
n E / kT 1 ( T ) F 费米－狄拉克分布函数 e e n ( E ) E / kT 1 n e e 1 对自由电子，令 e F / kT , F : 化学势，Fermi 能量 e n (E)
F
1
e ( E EF ) / kT 1 1 F n ( F ) (化学势的分布 ) 2
Quantum Statistics
3. 玻色－爱因斯坦凝聚
Predicted 1924… …Created 1995
Q1: What Is Bose-Einstein Condensation?
Fermions (費米子) and Bosons (玻色子)
Not all particles can have BEC. This is related to the spin of the particles. The spin quantum number of a particle can be an integer or a half-integer. Single protons, neutrons and electrons have a spin of ½ . They are called fermions. They cannot appear in the same quantum state. BEC cannot take place. Some atoms contain an even number of fermions. They have a total spin of whole number. They are called bosons. Bosons show strong “social” behaviour, and can have BEC. Example: A 23Na atom has 11 protons, 12 neutrons and 11 electrons.
PnB 1 ( n 1)! Pnc1 PnB 1 (1 n) P1 PnB
再放一个的几率是相应经典几率的(n+1)倍，增强因子（1+n） 费米子：
(1 n) P1 , (n 0,1)
为得量子统计分布，还需精细平衡原理。
在处于平衡的体系中考虑两个能级i与j,上面分别有ni及nj个 粒子。令Ri->j代表从i到j的跃迁几率。
Cooling Down the Atoms
See the animation: http://www.colorado.edu/physics/2000/bec/what_is_it.html When the temperature is high, the atoms have high energies on average. The energy levels are almost continuous. It is sufficient to describe the system by classical physics. When the temperature is low, the atoms have low energies on average. It is necessary to describe the system by quantum physics. When the temperature is very low, a large fraction of atoms suddenly crash into the lowest energy state. This is called Bose-Einstein condensation.
:
B (1,2)
两个玻色子处于相同状 态的几率是经典几率的 倍 2 F (1,2) 0 Pauli不相容原理
推广到n个全同粒子（玻色子），n 个玻色子处于相同状态 的几率n！倍于其经典几率。
问题：将一个玻色子放在一个状态上去的几率和状态原有的玻 色子数目有何关系？
经典：在态上放 个粒子的几率 n 量子： PnB n! Pnc n! ( P1 ) n Pnc ( P1 ) n
3 2 EK EP KT
dN N 0 ( 2mkT ) e
dxdydzdp x dp y dp z
N 0：在E p 0处的单位体积内粒子数 E K : 动能；E p : 势能 对动量积分
Ep
1 2 EK ( p x p 2 p z2 ) y 2m y y dy , z z dz :
x x dx , dxdydz
dN ' N 0 e
kT
Ep
单位体积内的粒子数 N N 0 e
kT
玻尔兹曼分布
一个自由度为 的体系，其能量是广义 f 坐标、广义动量的函数 E E (q1 ,..., q f ; p1 ,..., p f ) 在相空间体积元 dq1 ...dq f dp1 ...dp f 中的粒子数是 d d (q1 ,...q f ; p1 ,..., p f ) Ce

Bose and Einstein
• In 1924 an Indian physicist named Bose studied the quantum behaviour of a collection of photons. • Bose sent his work to Einstein, who realized that it was important. • Einstein generalized the idea to atoms, considering them as quantum particles with mass. • Einstein found that when the temperature is high, they behave like ordinary gases. • However, when the temperature is very low, they will gather together at the lowest quantum state. This is called Bose-Einstein condensation.