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Flashcards in Chp1 Deck (24)
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1

P(particle in [a,b])

integral a-b of mod(phi(x))^2

2

DEfn: normalised

integral mod(phi(x))^2 =1

3

Defn: position Operator: x~=

x

4

Defn: Momentum Operator: p~

-ih(d/dx) (partial)

5

Defn: Energy Operator: H=

p~^2/2m + V(x~)=-h^2(d^2/dx^2)phi+V(x)phi(x)

6

DEfn: Time-dep SE:

ih(dPHI/dt)=H(PHI)
ih(dPHI/dt)=-(h^2/2m)(d^2/dx^2)PHI +V(x)PHI

7

Defn: Stationary state: Phi(x,t)=

phi(x)exp(-iEt/h)
phi(x) is an efn of H with eval E

8

Conservation eqn for P(x,t):

dP/dt=-dj/dx

9

j(x,t)=

-ih/2m(PHI*dPHI/dx-dPHI*/dx*PHI)

10

Defn: Inner Product
(phi,theta)

integral phi(x)*theta(x)dx

11

Defn: Norm of phi

(phi,phi)=integral mod(phi)^2 dx

12

Defn: phi=

(phi,Hphi)

13

Defn: Uncertainty
(Dx)^2=

)^2>=-^2

14

Defn: Q is Hermitian iff

(phi,Qtheta)=(Qphi,theta)
(Q real) x~,p~ and H all fit

15

Prop: Cauchy-Schwarz

norm(phi)norm(theta)>=mod((phi,theta))

16

Thm: Ehrenfest's

d/dt=1/m


d/dt

=-

17

Thm: Heisenberg uncertainty

(Dx)(Dp)>=h/2

18

Defn: Commutator: [Q,S]=

QS-SQ

19

[x~,p~]

ih

20

Defn: Wavepacket

A wavefn localised in space

21

Defn: Gaussian Wavepacket

PHI(x,t)=(a/pi)^(1/4)1/sqrt(g(t)) e^(-x^2/2g(t))

22

correct g(t) for G wavep to solve tdep SE

a+ih/m t

23

Thm: General Ehrenfest

ihd/dt= {+ih}

24

Defn: Degeneracy

The dimension of the espace of eval l