Using Dirac Notation

8/10/2019 Using Dirac Notation

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5.2 Functions and Dirac notation

Slides: Video 5.2.5 Using Dirac

notationText reference: Quantum Mechanics

for Scientists and EngineersSection 4.1 (remainder of 4.1)



Functions and Dirac notat

Using Dirac notation

Quantum mechanics for scientists and engineers Da



Bra-ket notation and expansions on basis

Suppose the function is not represented directly

as a set of values for each point in space

but is expanded in a complete orthonormal b

We could also write the function as the “ket”

(with possibly an infinite number of elements)In this case, the “bra” version becomes

n n

n

f x c x

f

1 2 3 f x c c c




When we write the function in this different form

as a vector containing these expansion coeffic

we say we have changed its “representationThe function is still the same function

the vector is the same vector in our spWe have just changed the axes we use to represe

functionso the coordinates of the vector have changed

now they are the numbers c1, c2 , c3

f x

f x




Just as before, we could evaluate

so the answer is the same no matter how we w

2

, ,

1

21 2 3

3

n n m

n m

n m n m n m nm

n m n m

f x dx f x f x dx c x c

c c x x dx c c

c

cc c c f x f xc



Bra-ket notation and expansions on b

Similarly, with

we have

n n

n

g x d x

1

2

1 2 3

3

c

cg x f x dx d d d

c

g x f x



Bra-ket expressions

Note that the result of a bra-ket expression like

or

is simply a number (in general, complex)

which is easy to see if we think of this as a v

multiplicationNote that this number is not changed as we chan

representation just as the dot product of two vectors

is independent of the coordinate system

f x f x g x f x



Expansion coefficients

Evaluating the cn in

or the d n in

is simple because the functions are or

Since is just a functionwe can also write it as a ket

To evaluate the coefficient cm

we premultiply by the bra to get

n n

n

f x c x

n nn

g x d x n x

n x

m

m n m n n mn

n n x f x c x x c

n



Expansion coefficients

Using bra-ket notation

we can write as

Because cn is just a numberit can be moved about in the product

Multiplication of vectors and numbers is com

Often in using the bra-ket notation

we may drop arguments like x

Then we can write

n n

n

f x c x n n n n n n

n n n

f x c x x c x

n n n n

n n n

f c c



State vectors

In quantum mechanics

where the function f represents the state of the

mechanical systemsuch as the wavefunction

the set of numbers represented by the brket vector

represents the state of the system

Hence we refer to

as the “state vector” of the system

and as the (Hermitian) adjoint of the sta

f

f

f



State vectors

In quantum mechanics

the bra or ket always represents either

the quantum mechanical state of thesystem

such as the spatial wavefunctionor some state the system could be in

such as one of the basis states

n x



Convention for symbols in bra and ket vec

The convention for what is inside the bra or ket i

usually one deduces from the context what is

For exampleif it is obvious what basis we were working wit

we might use to represent the nth basis fbasis “state”)

rather than the notation orThe symbols inside the bra or ket should be enoumake it clear what state we are discussing

Otherwise there are essentially no rules for the

n

n x n



Convention for symbols in bra and ket vec

For example, we could write

but since we likely already know we are discus

such a harmonic oscillatorit will save us time and space simply to write

with 0 representing the quantum number

Either would be correct mathematically

20.375

The state where the electron has the lowest

possible energy in a harmonic oscillator with

potential energy x



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Using Dirac Notation