Chp1

Question 1

P(particle in [a,b])

Answer 1

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

Question 2

DEfn: normalised

Answer 2

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

Question 3

Defn: position Operator: x~=

Answer 3

x

Question 4

Defn: Momentum Operator: p~

Answer 4

-ih(d/dx) (partial)

Question 5

Defn: Energy Operator: H=

Answer 5

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

Question 6

DEfn: Time-dep SE:

Answer 6

ih(dPHI/dt)=H(PHI)

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

Question 7

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

Answer 7

phi(x)exp(-iEt/h)

phi(x) is an efn of H with eval E

Question 8

Conservation eqn for P(x,t):

Answer 8

dP/dt=-dj/dx

Question 9

j(x,t)=

Answer 9

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

Question 10

Defn: Inner Product

phi,theta

Answer 10

integral phi(x)*theta(x)dx

Question 11

Defn: Norm of phi

Answer 11

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

Question 12

Defn: phi=

Answer 12

(phi,Hphi)

Question 13

Defn: Uncertainty

(Dx)^2=

Answer 13

)^2>=-^2

Question 14

Defn: Q is Hermitian iff

Answer 14

(phi,Qtheta)=(Qphi,theta)

(Q real) x~,p~ and H all fit

Question 15

Prop: Cauchy-Schwarz

Answer 15

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

Question 16

Thm: Ehrenfest’s

Answer 16

d/dt=1/m<p>

d/dt</p><p>=-</p>

Question 17

Thm: Heisenberg uncertainty

Answer 17

(Dx)(Dp)>=h/2

Question 18

Defn: Commutator: [Q,S]=

Answer 18

QS-SQ

Question 19

[x~,p~]

Answer 19

ih

Question 20

Defn: Wavepacket

Answer 20

A wavefn localised in space

Question 21

Defn: Gaussian Wavepacket

Answer 21

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

Question 22

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

Answer 22

a+ih/m t

Question 23

Thm: General Ehrenfest

Answer 23

ihd/dt= {+ih}

Question 24

Defn: Degeneracy

Answer 24

The dimension of the espace of eval l