knl2 (r) = 2m(enl U (r))

3.2. TRAPPED NON-INTERACTING FERMI GAS AT T=0

55

3.2Trapped non-interacting Fermi gas at T=0

We consider the isotropic case (U (r)) and large number of particle (N 1). The energy depends only on the quantum numbers n and l and does not depend on the projection of the angular momentum, i.e.

	en = enl	3			(3.70)
	en = ~w(2n + l +		) for UI	0	(3.71)
		2

Figure 3.3: Schematic view of a trapped particle with large n
Since the system is rotational invariant, we have
y		=	cnl	Y (~r)	(3.72)
	nlm		r lm

Here Ylm(~r) are the spherical harmonic functions and c obeys the radial SCHRÖ - DINGER equation

d2 c	nl	+ knl2	(r)c		(r) = 0	with	(3.73)
				nl
dr2

~2l(l + 1)

r2

(3.74)

Since N 1 we know that only particles with n 1, i.e. those near the FERMI surface, can be excited. We can therefore apply WKB approximation.

If we call the classical turning points r1 and r2 we can approximate the radial wave function as

cnl

r2

p

cnl (r) =

cos

Zr1

dr0 pnl (r0)

for r1 < r < r2

(3.75)

p

4

pnl

pnl (r) = s

(3.76)

2m(enl

U (r)) ~2 (

r22 )

l + 1 2

56	CHAPTER 3. FERMIONS

This regime (n 1 but l arbitrary) is called THOMAS-FERMI regime. To calculate the density profile we use the semi-classical B OHR quantitazion requirement

r2	dr0 pnl (r0) = p~ n +	1
Zr1				(3.77)
		2

First we want to calculate the normalization coefficient cnl :

3

2

cnl

2

2

∞

2

Z

d

r jynlmj

= 1 = Z

dr

j j

r

= Z0

dr jcj

thus

(3.78)

r2

1

r2

c2

f:::g

(3.79)

= Zr1

dr pnl (r) cos

2

nl

r2

1

1

= cnl2

Zr1

dr

(1

+ cos[2 f:::g])

(3.80)

pnl

2

c2

r2

1

nl

Zr1

dr

(3.81)

2

pnl

Since the integral over strongly oscillating terms almost vanishes we can neglect the second term in (3.80). To calculate the other term we differentiate (3.77) in respect to n. Since the integrand pnl (r) vanishes at the limits of the integration we only have to differentiate the integrand, i.e. differentiating in respect to the upper bound gives

¶r2 ¶

r2

¶r2

Zr1

dr pnl (r) =

pnl

(r2) = 0

(3.82)

¶n

¶r2

¶n

Therefore we have

¶enl

r1

dr

p~ = m

Zr2

(3.83)

¶n

r2m(enl U (r))

~2

(l+ 1 )2

2

r2

¶enl

r2

1

= m

Zr2

dr

(3.84)

¶n

pnl (r)

)

cnl2

=

m ¶enl

(3.85)

2

p~

¶n

Note that the normalization coefficient is formally independent of the potential (which is of course relevant through the energy eigenvalues e).

To calculate n we note, that

l			2l + 1
å jYlm(~r)j2
			=							(3.86)
					4p
m= l
	¶en					¶e
ån		f (en) = Z			dn			f (e) = Z	de f (e)	(3.87)
	¶n					¶n

3.2. TRAPPED NON-INTERACTING FERMI GAS AT T=0

Now the particle density profile is

n(r) = g åjyn (~r)j2

n

2

1

cnl2

2

= g å jYlm(~r)j

cos

fg

r2

p

nl

nlm

g

ånl

2l + 1

cnl2

1

4p

r2

pnl (r)

2

=

g

ånl

2l + 1 m ¶enl

1

4p

r2

p~

¶n

r2m(enl U (r)) ~2

(l+ 21 )2

r2

m

å

2l + 1

å

¶e

1

= g

nl

4p~

r2

¶n

p

:::

l

n

eF

m

2l + 1

1

= g

ål

Zel (r) de p

4p~

r2

:::

eF

= g 4p~

ål

r2

s2m(enl U ) ~2

r22

2

1

2l + 1

l + 1

el (r)

57

(3.88)

(3.89)

(3.90)

(3.91)

(3.92)

(3.93)

(3.94)

= g

4p~ ål

r2

s

2m(eF U ) ~2

r22

(3.95)

1

2l + 1

l +

1

2

= g

1

Z0xmax(r) dx p

(3.96)

2m(eF U (r)) x

4p~3

= g		1			2	(2meF	3		( )	)	3	xmax(r)	(3.97)
		4p~3			3							0
=		g		(2m(eF U (r)))				U r		x	2		(3.98)
								2
	6p2~2

at T = 0. The summation/integration limits have always to be chosen such that p remains real.

This result means that if we redefine

p2F(r) +U (r) = eF

2m

we get locally the same result as in the non trapped case

p3 (r) n(r) = g F

6p2~3

but now with local FERMI momentum.

(3.99)

(3.100)