1b 61 h Ñ 0 - nankai university

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Nankai University Nankai University

H

h gt F kx E mv

M zM

Nankai University

x xxx- xx

xd dxx

c xccx

x

dx

x xxx- xx

xd dxx

c xccx

x xxx- xx

xd dxx

c xccx

x

dx

D x

Df x f xx xD x e x e

Nankai University

A B f Af Bf

D x

D x D x x x x

ABf A Bf

Df x Df x f x f x

 f

Â

Af Bf

A B

B

B

Nankai University

Dxf x xf x f x xf xx

xDf x xf xDx I xD

Î unit operator

null operator

AB BA

  Â

D f x D Df Df f f xx

Dx

Nankai University

A BC AB C

A x B x CAB Dx xD BC x

commutator A B AB BA

AB BA A B A B

AB BA A B

AB C f xD f f xf

A BC f D xf f xf

Nankai University

i j ji

A p i B q qq

i j j ij ji i

AB p q i q i qq q

i ji j i j j i

ij i j

i j p qp q p q q p

i i j p q

ij

i ji j

x x xx

j i ji

BA q p i qq

i j j i ijp q q p i

Nankai University

 Â

A c f c g c Af c Ag

c c f g

c f c g c f c g

c c c c c c

d dx x x y

Nankai University

A B C AC BC

A B C AB AC

A B C f A B Cf

A Cf B Cf

ACf BCf

AC BC f

A B C

A B C f AC BC f

A B C AC BC

Nankai University

D x D x D x

D D x x D x

D Dx xD x

D xD x

x x f

D x f D x D x f

D x f xff f xf xf x ff xf x f

D xD x f

D x D xD x

Nankai University

 aA a

 a

ax axe aex

eax x a

ax axe axex

eax x

x x

x x xx x

Nankai University

x

f x kf xx

f k xf

f kx

kxf Ce k

f  a

cf c  a

Nankai University

A A d

A A

A A d A A d

A d A d

Â

f g

f Agd g Af d

Nankai University

f Cg C

A d A d

f Cg A f Cg d f Cg A f Cg d

f C g Afd f C g ACgd

f Cg Af d f Cg ACg d

f Afd C g Afd C f A CC g Ad dg g

f Af d C g Af d C f Ag CC g Agd d

C C

C i

g Afd f Agd g Af d f Ag d

g Afd f Agd g Af d f Ag d

f Agd g Af d

Nankai University

A a A a

f Agd f A g g Af d g A f

m n m nA d A m A n

m nd m n

m n m n

Nankai University

A a A a

A d a d a d

A d a d a d

a a

A d i A i A d i A i

i A i i A i

i a i i a i

a i i a i i a i i

a a i i

a a

Nankai University

m n  am anam an

 m= am m  n= an n

 m am  n an n

m  n n  m

m an n n am m

an m n am n m am m n am m n

am an m nam an m n

Nankai University

 n1 n2 nfan an f f

n1 n2 nf ff nj

f

nj ji nii

c j f

 f =af Âg= ag f g

f g cf

f g+cf f g c f f f gdf g

cf f f fd

Nankai University

n x

f x xa b

i ii

f x c x i

n* x x a b

b b

n i n i i ni ni ia a

x f x dx c x x dx c c

Nankai University

ÂÂ

i i =si i

i =ti i , f

i ii

f c i ii

i ii

i i ii

i i i i ii

i i i i i i ii

AC CA f AC CA c

c AC CA

c A C C A

c At Cs

c s t t s

Nankai University

i i =ci i ,

i i iAC Ac

i i iC A c A

i i cici

i =ai i

ci n i ci nn

n

Nankai University

fi fj ai aj ai aj

i j

ai fi fi ci fi

i j cj i j

ai n Âfk ai fk k nn

i k kk

Cf c f

n ni k k kk k

i C j i Cj f c f c i j

Nankai University

q tq t q q qf

t f = 3N N

* d d t ft q q q dq q q q dq qf

qf qf dqf

Nankai University

iEtq t e q

Nankai University

*

x

x

x

x

Nankai University

x

x

x

x

Cd

KdK

dK

d

k

Nankai University

e x e x e x|

e x

x

e |x|

x

e x

x

e x x e x

e x

e x| x

Nankai UniversityNankai University

F q t FtqFtqF

tq

iq

iqqGtppqqGf

fff

q p t p

Nankai University

ii q

ipy

ipy

xipx

zipz

m V x y zT

zyx pppm

T

zyxmppp

mT zyx

zyxVzyxV

zyxVzyxm

H

E T V

Nankai University

xyzxyz

zyx

ypxpkxpzpjzpypipppzyxkji

prM

Mx yp zp

My p p

Mz p p

yz

zyiM x

zx

xziM y

xy

yxiM z

M Mx My Mz

xy

yx

zx

xz

yz

zyM

Nankai University

A  q ta q t  q t a q t

A q t a a Âq t Â

 q t a q t Â

q t t tAdAA

Nankai University

  Â

i j i j j i

i j j i

j i j i j i

ij

p q p q q p

i q q q i q

q i q i q q q i q

i

i ip q i

ii

p iq

x

px

y

pypz

z

Nankai UniversityNankai University

H it

E t

E i t

E t Et tE i

Nankai University

q t q f tf tH f t H q i i q

t tH q f ti

q f t tE

H q E q

df t iE dtf t

iEtf t Ae

iEtq t e q

Nankai UniversityNankai University

E ÂH A d HA AH

H A E A

H A A H

E A E A

H A d

hypervirial theorem

i i ii i i

A q p i qq

i i i i i i i ii i i i

H q p H q p q H p H q p

Nankai University

x y z

xx x

H x T V x T x V x y z x

p x p x p xm m m

pp x xp im m

x y zH T V p p p V x y zm

x x x xVH p T V p T p V x y z p ix

i i i i i ii i i

ii

i ii

ii i

H q p q H p H q p

pVi q iq mVi q i Tq

ii i

Vq T Tq

virial theorem

Nankai University

f x x xj f sx sx sxj sn f x x xj

f n nN

ii i

fx nfx

V n

T n V

T V E

N

ii i

Vq nVq

EVn

nETn

Nankai University

A B B A

nA A

A c

cA B A cB c A B

A B C A B A C

A B C A C B C

A BC A B C B A C

AB C A B C A C B

A B C B A C C A B

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