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Too Many to Count. Three Notations. The Three Notations of Quantum Mechanics. There are three notations (dialects if you like) commonly used in quantum mechanics Sometimes they can be used interchangeably and sometimes not Each has a strength and each has a weakness

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The three notations of quantum mechanics l.jpg
The Three Notations of Quantum Mechanics

  • There are three notations (dialects if you like) commonly used in quantum mechanics

  • Sometimes they can be used interchangeably and sometimes not

    • Each has a strength and each has a weakness

  • They are named for the 3 “fathers” of quantum mechanics

    • Schroedinger

    • Heisenberg

    • Dirac



Postulate 1 l.jpg

Quantum Mechanical States are described by vectors in a linear vector space

Linear vector space means a field of scalars over which the space

From section 4.4 of Liboff’s text

Postulate 1


Actually this is nothing new l.jpg
Actually this is nothing new linear vector space


Postulate 2 l.jpg

A dual space exists with the same dimensionally as the original vector space

AKA “dual continuum”

the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1)

Required to allow the inner product so that vectors can be normalized

Postulate 2


Dual spaces for the notations l.jpg
Dual Spaces for the notations original vector space

  • To transform a vector from one space to another, a Hermitian conjugation is performed.


Postulate 3 l.jpg

An inner product exists. original vector space

Back in E&M, we called the inner product: “dot product”

Inner product = dot product = scalar product

Postulate 3


In the 3 notations l.jpg
In the 3 notations original vector space


Postulate 4 l.jpg
Postulate 4 original vector space

The dual space is linear and has the following property


Postulate 5 l.jpg
Postulate 5 original vector space


Postulate 6 l.jpg
Postulate 6 original vector space


Postulate 7 l.jpg

Multiplying a ket by a complex number (different from zero) does not change the physical state to which the ket corresponds

Postulate 7


Postulate 7 is discussing normalization l.jpg
Postulate 7 is discussing normalization does not change the physical state to which the ket corresponds


It is convenient for define an orthonormal basis and you ve been doing it all your life l.jpg
It is convenient for define an orthonormal basis (and you’ve been doing it all your life!)


Operators l.jpg
Operators you’ve been doing it all your life!)

  • A mathematical operation on a vector which changes that vector into another

    • This is not mere multiplication (like Postulate 7) but we are actually changing something like its direction or perhaps other quantities.

    • Example: Let Q be the differential operator with respect to x

Direction of operation

Direction of operation


Postulate 8 l.jpg

Physical observables (such as position or momentum) are represented by linear Hermitian operators

Postulate 8


What does linear mean l.jpg
What does linear mean? represented by linear Hermitian operators


What does hermitian mean l.jpg
What does Hermitian mean? represented by linear Hermitian operators


A special case for operators l.jpg
A special case for operators represented by linear Hermitian operators

Called

“Eigenvector” or

“Eigenfunction” or

“Eigenket”

Called “eigenvalue”


What does an eigenvalue mean in schroedinger notation l.jpg
What does an eigenvalue mean in Schroedinger notation? represented by linear Hermitian operators


What does an eigenvalue mean in heisenberg notation l.jpg
What does an eigenvalue mean in Heisenberg notation? represented by linear Hermitian operators


Theorem 1 l.jpg

Eigenvalues of a Hermitian operator are real represented by linear Hermitian operators

i.e.

If Q+=Q then q*=q

Theorem 1


Proof of thm 1 l.jpg
Proof of Thm 1 represented by linear Hermitian operators


Theorem 2 l.jpg

Eigenvectors of a Hermitian operator are orthogonal if they belong to different eigenvalues

Theorem 2


Proof of thm 2 l.jpg
Proof of Thm 2 belong to different eigenvalues

Note: An operator may have a set of eigenvalues of which 2 or more are equal; this is called degeneracy


Projection operators l.jpg
Projection operators belong to different eigenvalues

  • Graphically, the inner product represents the project of a onto b or in Dirac notation |a> onto |b>

|b>

|a>

<a|b>

If |a> is considered a unit vector, then the vector which represents projection of |b> onto |a> is written <a|b>|a> or |a><a|b>


Theorem 3 l.jpg
Theorem 3 belong to different eigenvalues

A projection operator is idempotent i.e.

Q2 =Q


Theorem 4 l.jpg
Theorem 4 belong to different eigenvalues


Proof of thm 4 l.jpg
Proof of Thm 4 belong to different eigenvalues


Creating a set of orthogonal vectors from a set of normalized linear independent kets l.jpg
Creating a set of orthogonal vectors from a set of normalized linear independent kets

  • Let |a>, |b>, and |c> be a set of normalized linear independent kets

  • We are going to create a new set of kets (|1>, |2>, |3>) from these which will be orthogonal to one another i.e. <1|2>=0, <1|3>=0 and <2|3>=0

  • First, pick one of the original set and build the rest of the set around it

  • |1>=|a>


Constructing 2 l.jpg
Constructing |2> normalized linear independent kets

Geometrically

|2>=|b>-|1><1|b>

|b>

|b>-|1><1|b>

|1><1|b>

|1>

-|1><1|b>


Test that 2 is orthogonal to 1 l.jpg
Test that |2> is orthogonal to |1> normalized linear independent kets


Normalizing 2 l.jpg
Normalizing |2> normalized linear independent kets


Slide36 l.jpg
|3> normalized linear independent kets


Postulate 9 l.jpg

Eigenvalues are the only possible outcome of physical measurements

If physical observables are represented by Hermitian operators and these have real eigenvalues, it is reasonable to assume that there is a connection between their eigenvalues and the results of experiments.

Postulate 9



Proof of thm 5 l.jpg
Proof of Thm 5 commute

Commutator Brackets

[a,b]=(ab-ba)

If [a,b]=0 then a and b commute

QM analog of Poisson brackets



Postulate 10 l.jpg
Postulate 10 commute

The average value in the state |a> of an observable represented by an operator Q, is

Called an “expectation value” or called the “mean”




Defining standard deviation l.jpg
Defining Standard Deviation commute

  • Let Q= operator

    • DQ= standard deviation of measurement of Q

    • (DQ)2= variance of that measurement

      • Sometimes called mean square deviation from the mean

      • (DQ)2 =<(Q-<Q>)2>

      • Or, more compactly

      • (DQ)2 =<Q2>-<Q>2


The uncertainty principle l.jpg
The Uncertainty Principle commute

  • If two observables are represented by commuting operators then you can measure the physical observables simultaneously

  • If the operators DO NOT COMMUTE then a SIMULTANEOUS measurement will NOT BE EXACTLY REPEATABLE

  • There will be a spread in the measurement such that the product of the standard deviations will exceed a minimum value; the size of the minimum depends on the observable

  • To calculate this, we first have to build some mathematical machinery.


Theorem 6 l.jpg
Theorem 6 commute

Schwartz’s Inequality



Theorem 7 l.jpg
Theorem 7 commute

Let a = A-<A> and

b =B -<B> then

[a,b] =[A,B]




Need more power l.jpg
Need more power! commute

  • Now the absolute square of any complex number, z, can be written as

    • |z|2 = (Re(z))2 +(Im(z))2

    • Of course, |z|2 (Im(z))2


An aside l.jpg
An Aside commute




Does it work l.jpg
Does it work? commute


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