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Summary Quantum Mechanics II (8.05) - 2008 Review Notes
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Work at MIT Quantum Mechanics II (8.05): review notes for the final exam.
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Quantum Review Notes (8.05) Page 1 of 128.05 Review Notes
Information on the formula sheet is not usually
reproduced here…
The First Bits of the Course…
For a free particle
ˆ pˆ2 2 ¶2
H = +V (x ) = +V (x )
2m 2m ¶x 2
General features of wavefunctions
o The ground state must be even.
o The number of nodes indicates how “high” the state is.
To incorporate the fact a particle decays as exp (-t / t ) , add -i / 2t to
the potential.
The particle flux is given by J (x , t ) =
m (
Im y * ¶y
¶x ) with r + J ¢ = 0 . To
prove, write an expression for r = d
dt
(yy * ) and simplify with a complex-
conjugated SE. Integrating the conservation law over all space, we end up
with the fact total probability is conserved. For a fluid, J = rv which
gives us a nice definition of quantum velocity.
éxˆ, pˆù = i
êë úû
General tips and tricks with Dirac Notation
o A crucial step in many derivations is that
x | pˆ | p = p x p
1
=p e ipx /
2p
d 1
= -i e ipx /
dx 2p
d
= -i x p
dx
And similarly that
© Daniel Guetta, 2008
, Quantum Review Notes (8.05) Page 2 of 12
p | xˆ | x = x p x
1
=x e -ipx /
2p
d
= i x p
dp
o When working out expressions like x | pˆ | y , write it as
ò x | pˆ | p p | y dp .
o To find x | pˆ | y , insert the identity into x | pˆ | y and compare.
When showing that e  a is an eigenstate (of x, say), easiest way to do it
is
ˆ ˆ ˆ
ˆ A a = e A e -Axe
xe (ˆ
)
ˆ A a = e A xˆ + éêxˆ, A
ˆ
ë (
ˆù a
úû )
ˆ
[This uses the fact that e ABe
ˆ
ˆ, Bˆ ] , quoted on the formula sheet].
ˆ -A = Bˆ + [A
To find the Fourier Transform of a function like xeipx, eliminate the x by
expressing it as a derivative of the exponential.
The function of the doubly differentiated delta function is to “pick out”
the double derivative.
To find the maximum and/or minimum value of an operator  , consider a
ˆ | y = a y y . Then,
normalised eigenvector y and realise that y | A
ˆ | y a norm, and realise it must
write  in two ways that makes y | A
therefore be greater than 0 (for example, y | aˆ†aˆ | y = aˆ y ³ 0 and
ˆˆ† | y ).
y | 1 - aa
For a free particle, the wavelength is given by
2k 2
= E -Veff
2m
Translation operators in QM
x ˆ y = y(x - x 0 ) ˆ†xˆˆ = x + x 0
-ix 0 pˆ/
=e
i fJˆz
=e rotates the system by f about the z-axis
The postulates of QM:
o At each instant, the state of a physical system is represented by a
ket y in the space of states.
© Daniel Guetta, 2008
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