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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

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 1A 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 2Dual Spaces for the notations

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

An inner product exists.

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

Inner product = dot product = scalar product

Postulate 3Postulate 4

The dual space is linear and has the following property

Multiplying a ket by a complex number (different from zero) does not change the physical state to which the ket correspondsPostulate 7

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

Operators

- 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

Physical observables (such as position or momentum) are represented by linear Hermitian operatorsPostulate 8

Eigenvectors of a Hermitian operator are orthogonal if they belong to different eigenvaluesTheorem 2

Proof of Thm 2

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

Projection operators

- 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>

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>

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 9Proof of Thm 5

Commutator Brackets

[a,b]=(ab-ba)

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

QM analog of Poisson brackets

Postulate 10

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

- 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

- 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

Schwartz’s Inequality

Need more power!

- 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

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