Pessue in the Aveage-Atom Moe W.. Johnson Depatment of Physics, 225 Nieuwan Science Ha Note Dame Univesity, Note Dame, IN 46556 Febuay 28, 2002 Abstact The (we-known) quantum mechanica expession fo the stess tenso is eive an appie to obtain a fomua fo the pessue in the aveage-atom moe. This aveage-atom pessue fomua euces to the (we-known) expession fo the pessue in a cassica fee-eecton gas when the aveage-atom continuum wave functions ae epace by fee-eecton wave functions. 1 Deivation We stat with the time-epenent Schöinge equation fo an eecton in a potentia V (), h i t = h2 2m 2 ψ Vψ (1) The expectation vaue of i-th component of the eecton s momentum insie a egion is p i = τ ψ p i ψ. (2) The ate of incease of momentum in is [ t p i = τ t p iψ ψ p i t = i h [ τ 2 ψ p i ψ ψ p i 2 ψ 2m ī τ ψ [p i V Vp i ψ. (3) h 1
This expession can be ewitten as t p i = i h [ τ ψ p i ψ ψ p i ψ 2m ī τ ψ [p i,v ψ. (4) h With the ai of Gauss theoem, Eq. (4) euces to: t p i = i h 2m S = h2 2m S [ n p i ψ ψ p i ī τ ψ [p i,v ψ x x h n [ x ψ 2 ψ x τ ψ V ψ (5) The fist intega is the i-th component of the suface foce on the egion an the secon gives the i-th component of the voume foce. We intouce the stess-tenso [ T i = h2 ψ 2 ψ (6) 2m x x an the voume foce V F i =. We fin that the time ae of change of momentum is t p i = S T i n F i. (7) Fom this expession it foows that T in is the i-th component of the foce pe unit aea exete by the suounings on the egion though the suface. Theefoe T i is the i-th component of the foce/aea, on a suface with noma in iection n exete by the eectons in the egion on the suounings. The pessue is eate to the tace of the stess tenso by P = 1 T ii. (8) 3 In the stationay state, we must have S T i n = F i, (9) i 2
which euces to T i x = ψ ψ V (10) in iffeentia fom. It is not ifficut to veify the iffeentia fom of the momentum consevation aw above iecty fom the singe-patice Schöinge equation. We stat with the equation fo / We eft mutipy this by ψ to obtain h2 2 =(E V ) V ψ. (11) 2m h2 2m ψ 2 =(E V ) ψ ψ V ψ. (12) We next consie the equation fo ψ ight mutipie by /. h2 2m 2 ψ =(E V ) ψ. (13) Subtacting (13) fom (12), one obtains h 2 [ 2 ψ ψ 2 = ψ V ψ. (14) 2m This equatiom may be simpife to h 2 [ 2m ψ ψ = ψ V ψ. (15) Setting [ T i = h2 2m x we see that Eq. (15) becomes T i x ψ = ψ ψ V, 2 ψ x which is pecisey the iffeentia fom of the momentum consevation aw given eaie in Eq. (10)., 3
2 Evauation of Pessue We fist evauate the fomua fo pessue given in Eq. (8) fo an eecton in state (nm) with wave function ψ nm () = 1 P n() Y m (ˆ). Utimatey, we sum the eecton patia pessues ove cose subshes. Fo one eecton, we have We note that Thus ψ nm () = P = h2 6m ( ) Pn () [ ψ ψ nm ψ nm = [ ( Pn () ( ) ( 1) P n () Pn () ( ( 1) Pn () 2 Futhemoe, we have ψ nm 2 ψ nm = P n () [ 1 2 2 2.1 Usefu Ientities ( Pn () ψ ψ 2 ψ Y ( 1) m (ˆ)P n() ( 1) ) 2 ( 1) m Y ( 1) m ( 1) m [ Y ( 1) m ) ( 1) (ˆ) Y m (ˆ) (16) Y (1) m (ˆ). (17) (1) (1) ( 1) (ˆ) Y m (ˆ)Y m (ˆ) Y m (ˆ) ) 2 ( 1) m Y (1) m ( 1) 2 (1) (ˆ) Y (ˆ). (18) m P n () ( 1) m Y m (ˆ)Y m (ˆ) (19) One may easiy estabish the foowing theoem: ( 1) m Y m (ˆ)Y m (ˆ) = [ 4π. (20) m 4
We expan the vecto hamonics as Y (1) JM (ˆ) = J 1 Y JJ 1M (ˆ) [J Y (0) JM (ˆ) = Y ( 1) JM (ˆ) = J 1 [J J [J Y JJ1M(ˆ) (21) Y JJM (ˆ) (22) J [J Y JJ 1M(ˆ) J 1 [J We can pove by iagammatic methos that ( 1) M Y JK M (ˆ) Y JLM (ˆ) = (1) M M Y JJ1M (ˆ). (23) JL1 [J 4π δ KL. (24) With the ai of this esut, it foows that ( 1) M Y (λ) (µ) [J J M (ˆ) Y JM (ˆ) =( 1)λ1 4π δ λµ. (25) 2.2 Summay Combining Eqs. (18) an (19), we fin [ ψ nm ψ nm ψ nm 2 ψ nm = m [ 4π { [ ( ) Pn () 2 P n() 2 2 P n () 2( 1) 2 2 ( ) } Pn () 2. (26) The patia pessue fom a cose subshe n point may, theefoe, be witten { [ ( ) } P = h2 2[ 6m 4π 2 2 Pn () 2 ( 1) 2 Pn 2 ()2m h 2 (E n V ()) Pn 2 (). (27) If we choose to be the aius of the aveage atom V () = 0 then { [ ( ) P = h2 2[ Pn () 2 ( ) ( 1) Pn () 2 6m 4π 2 2m ( ) } h 2 E Pn () 2 n. (28) 5
Thee ae two contibutions to the pessue at the suface of the aveage atom sphee: P boun = 1 2(2 1) 24πm 1e (ɛ n µ)/kt n { [ ( ) Pn () 2 ( ) ( 1) Pn () 2 2 2m ( ) } h 2 E Pn () 2 n (29) P contin = 1 ɛ 24πm 0 1e (ɛ µ)/kt 2(2 1) { [ ( ) Pn () 2 ( ) ( 1) Pn () 2 ( ) } 2 p 2 Pn () 2 (30) 2.3 Fee Eecton Gas Fo a fee eecton gas, P ɛ () = 2m πp p (p). The coesponing pessue at the suface of the aveage atom sphee is P fee = h2 ɛ 2m 24πm 0 1e (ɛ µ)/kt πp p4 { ( ) (z) 2 2(2 1) ( 1) z 2 2 (z)2 (z) } z=p (31) Now, we state a few usefu theoems: 1. Fist we use Eq. (10.1.50) in [1 (2 1) 2 (z) =1. 2. Diffeentiating with espect to z gives (2 1) (z) (z) =0. 6
3. Diffeentiating once again, one fins ( ) (z) 2 (2 1) = (2 1) (z) 2 (z) 2. 4. Substituting fom the iffeentia equation fo spheica Besse functions, ( ) (z) 2 (2 1) = [ 2 (2 1) z (z) ( ) (z) ( 1) 1 z 2 2 (z) = ( ) ( 1) (2 1) 1 z 2 2 (z) 5. Fom this, it foows that { ( ) } (z) 2 ( 1) 2(2 1) z 2 2 (z)2 (z) = 4 (2 1) 2 (z) =4. (32) With the ai of Eq. (32), we we may ewite the expession fo the pessue as P fee = (2m)3/2 ɛ 3/2 ɛ 3π 2 (2mkT )5/2 = 6mπ 2 0 1e (ɛ µ)/kt 0 y 3/2 y 1e (y x) (2mkT )5/2 = 6mπ 2 I 3/2 (x) (33) whee x = kt. This expession agees with the cassica expession fo the pessue of a fee eecton gas given, fo exampe, in Feynman et a. [2 efeences [1 M. Abamowitz an I. A. Stegun, es., Hanbook of Mathematica Functions, Appie Mathematics Seies 55 (U. S. Govenment Pinting Office, Washington D. C., 1964). [2. P. Feynman, N. Metopois, an E. Tee, Phys. ev. 75, 1561 (1949). 7