Using Unitary Operators

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6.2 Unitary and Hermitian operators

Slides: Video 6.2.1 Using unitaryoperators

Text reference: Quantum Mechanicsfor Scientists and Engineers

Section 4.10 (starting from“Changing the representation ofvectors”)

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Unitary and Hermitian operato

Using unitary operators

Quantum mechanics for scientists and engineers David M

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Unitary operators to change representations of vecto

Suppose that we have a vector (function)that is represented

when expressed as an expansion onthe functions

as the mathematical column vectorThese numbers c1, c2, c3, …

are the projections ofon the orthogonal coordinate axesin the vector space

labeled with , , …

old f

n

old f

old f

1 2 3

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Unitary operators to change representations of vecto

Suppose we want to represent this vector on a new setof orthogonal axes

which we will label , , …Changing the axes

which is equivalent to changing the basis set offunctions

does not change the vector we are representingbut it does changethe column of numbers used to represent th

vector

1 2 3

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Unitary operators to change representations of vecto

For example, suppose the original vectorwas actually the first basis vector in the old basis

Then in this new representationthe elements in the column of numbers

would be the projections of this vectoron the various new coordinate axes

each of which is simplySo under this coordinate transformationor change of basis

old f

1m 1

0

0

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Unitary operators to change representations of vecto

Writing similar transformations for each basis vectorwe get the correct transformation

if we define a matrix

where

and we define our new column of numbers

11 12 13

21 22 23

31 32 33ˆ

u u u

u u uU u u u

ij i ju

ˆnew old f U f

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Unitary operators to change representations of vecto

Note incidentally that hereand are the same vector in the vector sp

Only the representationthe coordinate axes

and, consequentlythe column of numbers

that have changednot the vector itself

old f new f

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Unitary operators to change representations of vecto

Now we can prove that is unitaryWriting the matrix multiplication in its sum form

sohence is unitary

since its Hermitian transpose is therefore itinverse

U

†ˆ ˆmi mj

ijm

U U u u

†

ˆ ˆ ˆU U I U

m i m jm

i m

i m m jm

ˆ

i j I i j

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Unitary operators to change representations of vecto

Hence any change in basiscan be implemented with a unitary operator

We can also say thatany such change in representation to a new

orthonormal basisis a unitary transform

Note also, incidentally, that

so the mathematical order of this multiplicationmakes no difference

†

† † †ˆ ˆ ˆ ˆ ˆUU U U I I

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Unitary operators to change representations of opera

Consider a number such aswhere vectors and and operator are arbitr

This result should not depend on the coordinate systemso the result in an “old” coordinate system

should be the same in a “new” coordinate systemthat is, we should have

Note the subscripts “new” and “old” refer to representnot the vectors (or operators) themselveswhich are not changed by change of representati

Only the numbers that represent them are chan

ˆg A f f g ˆ A

og

ˆ new new new g A f

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Unitary operators to change representations of opera

With unitary operator to go from “old” to “new” sywe can write

Since we believe also that

then we identify

or since

then

†ˆ ˆnew new new new new newg A f g A f

U

ˆ new new new old g A f g †ˆ ˆˆ ˆ

old newU A U

† † †ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆold new newUA U UU A UU A

†ˆ ˆˆ ˆnew old A UA U

† ˆˆ ˆ

old new old U g A U f old g U

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Unitary operators that change the state vector

For example, if the quantum mechanical state

is expanded on the basis to givethen

and if the particle is to be conservedthen this sum is retained as the quantum

mechanical system evolves in timeBut this is just the square of the vector l

Hence a unitary operator, which conserves lengthdescribes changes that conserve the particle

n n

2

1nn

a

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