4.3.5_slides Time-evolution and Hamiltonian
-
Upload
jacob-bains -
Category
Documents
-
view
216 -
download
0
Transcript of 4.3.5_slides Time-evolution and Hamiltonian
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 1/13
4.3 Measurement and expectationvalues
Slides: Video 4.3.5 Time evolution
and the HamiltonianText reference: Quantum Mechanics
for Scientists and EngineersSection 3.11
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 2/13
Measurement and expectation v
Time evolution and the Hamilt
Quantum mechanics for scientists and engineers Da
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 3/13
Time evolution and the Hamiltonian
Taking Schrödinger’s time dependentequation
and rewriting it as
and presuming does not dependexplicitly on time
i.e., the potential is constant
could we somehow legally write
ˆ , H t r
,t
t
r
ˆ H
V r
1
1
ˆ, exp
iH t t
r
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 4/13
Time evolution and the Hamiltonian
Certainly,
if the Hamiltonian operator here was replac
constant numberwe could perform such an integration of
to get
1 0
1 0
ˆ, exp ,
iH t t t t
r r
ˆ, ,t iH t t
rr
ˆ H
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 5/13
Time evolution and the Hamiltonian
If, with some careful definition, it was legal to do
then we would have an operator that
gives us the state at time t 1 directly from thaTo think about this “legality”
first we note that, because is a linear operatfor any number a
Since this works for any function
we can write as a shorthand
ˆ H
ˆ ˆ
, , H a t aH t
r r
,t r
ˆ ˆ Ha aH
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 6/13
Time evolution and the Hamiltonian
Next we have to define what we mean by an operaised to a power
By we meanSpecifically, for example, for the energy eigenfun
We can proceed inductively to define all higher
which will give, for the an energy eigenfunctio
2ˆ H 2ˆ ˆ ˆ, , H t H H t r r
2 ˆ ˆ ˆ ˆ ˆn n n n n n H H H H E E H r r r r
1ˆ ˆ ˆm m H H H
ˆ m m
n n n H E r r
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 7/13
Time evolution and the Hamiltonian
Now let us look at the time evolution of somewavefunction between times t 0 and t 1
Suppose the wavefunction at time t 0 iswhich we expand in the energy eigenfunctio
asThen we know
multiplying by the complex exponential factorthe time-evolution of each basis function
,t r
r
n n
na
r r
1 0
1, exp n
n n
n
iE t t
t a
r r
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 8/13
Time evolution and the Hamiltonian
In
noting that
we can write the exponentials as power seri
so
1 0
1, exp n
n n
n
iE t t t a
r r
2 3
exp 1
2! 3!
x x x x
2
1 0 1 0
1
1, 1
2!
n n
n
n
iE t t iE t t t a
r
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 9/13
Time evolution and the Hamiltonian
In
because we showed thatwe can substitute to obtain
1 0 1 0
1
1, 1 2!
n n
n
n
iE t t iE t t
t a
r
ˆ m m
n n n H E r r
2
1 0 1 0
1
ˆ ˆ1, 1
2!n
n
iH t t iH t t t a
r
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 10/13
Time evolution and the Hamiltonian
With
because the operator and all its powers commwith scalar quantities (numbers) we can rewrite
ˆ H
1 0 1 0
1
ˆ ˆ1, 1
2!
n
n
iH t t iH t t t a
r
2
1 0 1 0
1
2
1 0 1 0
ˆ ˆ1, 1
2!
ˆ ˆ11 2!
iH t t iH t t t
iH t t iH t t
r
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 11/13
Time evolution and the Hamiltonian
So, provided we define the exponential of the opin terms of a power series, i.e.,
then we can write our preceding expression as
1 0 1 0 1ˆ ˆ ˆ1
exp 12!
iH t t iH t t iH t
1 0
1 0
ˆ, exp ,iH t t
t t
r r
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 12/13
Time evolution and the Hamiltonian
Hence we have established that
there is a well-defined operator that
given the quantum mechanical wavefunctio“state” at time t 0
will tell us what the state is at a time t 1
1 0
1 0
ˆ
, exp ,
iH t t t t
r r
8/10/2019 4.3.5_slides Time-evolution and Hamiltonian
http://slidepdf.com/reader/full/435slides-time-evolution-and-hamiltonian 13/13