4.3.5_slides Time-evolution and Hamiltonian

8/10/2019 4.3.5_slides Time-evolution and Hamiltonian

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4.3 Measurement and expectationvalues

Slides: Video 4.3.5 Time evolution

and the HamiltonianText reference: Quantum Mechanics

for Scientists and EngineersSection 3.11



Measurement and expectation v

Time evolution and the Hamilt

Quantum mechanics for scientists and engineers Da



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




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




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




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




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




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




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




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




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




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



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4.3.5_slides Time-evolution and Hamiltonian