Postulates of Quantum Mechanics

Question 1

Representation of Systems

Classical Physics

Answer 1

-in classical physics particles can be specified by its coordinates and momentum:
(x, y, x, px, py, pz)
-the Hamiltonian (H(x,y,z,px,py,pz) ) is a function that fully describes the time evolution of the system
-this method fails in quantum mechanics since we cannot specify coordinates and momentum at the same time, as described by the Heisenberg Uncertainty Principle

Question 2

Representation of Systems

Quantum Mechanics

Answer 2

• to describe quantum systems we need a vector space with the following properties:
  1) a linear vector spaces where scalars are complex numbers
  2) an inner product that maps two vectors into a scalar
• a vector space that satisfies these properties is called Hilbert space

Question 3

Dual Space

Answer 3

• for every Hilbert space we can associate a dual Hilbert space
• every vector in the Hilbert space has maps to another vector in the dual space
• the inner product takes a vector in the Hilbert space and a vector in dual space and maps them to a complex number
• the dual Hilbert space can be thought of as a mirror image of the Hilbert space

Question 4

Hilbert Space

Dirac Notation

Answer 4

• vector is the Hilbert space are referred to as ket vectors or kets denoted: |v⟩
• vectors in the dual Hilbert space are referred to as bra vectors or bras denoted: ⟨u|
• the inner product is denoted: ⟨u|v⟩

Question 5

Kets and Bras

Answer 5

• every ket vector has a corresponding bra vector and vice versa: ⟨v| |v⟩
• more generally: a* ⟨u| + b* ⟨v| = a|u⟩ + b|v⟩
• where a* and b* are complex conjugates

Question 6

Norm of a Ket Vector

Answer 6

-the inner product of the ket with its own bra vector:
||v||² = ⟨v|v⟩
-the norm of a ket is always real and positive

Question 7

Normalising a Vector

Answer 7

-divide by the square root of the norm:

|v⟩ / √⟨v|v⟩

Question 8

Conjugate Symmetry

Answer 8

⟨u|v⟩ = ⟨v|u⟩*

-where * indicated the complex conjugate

Question 9

Orthonormal Basis

Answer 9

-a basis is orthonormal if for,
⟨vi | vj⟩ = 𝛿ij
𝛿ij = (1 for i=j AND 0, for i≠j)

Question 10

Representing a Vector in the Hilbert Space in a Basis

Answer 10

-any vector in Hilbert space can be written as a linear combination of basis vectors { |vi ⟩} :
|ψ⟩ = Σ ci * |vi ⟩
-the coefficients are given by the inner product:
ci = ⟨vi |ψ⟩
-in the given basis the ket vector |ψ⟩ becomes a column vector with entries ci

Question 11

Changing Basis

Answer 11

• when we change basis, the representation (column vector) will also change
• however the norm of the vector will not change under basis transformation

Question 12

Postulate 1

Answer 12

States of Quantum Systems

• the state of any quantum system is specified by |ψ⟩ , a vector belonging to the Hilbert space
• all information about the system is contained in |ψ⟩
• if |ψ1⟩ and |ψ2⟩ are two states of the system, any linear combination is also a state

Question 13

Representation in a Continuous Basis

Answer 13

-in some cases we need to allow for the possibility of an infinite basis
-we generalise the condition of orthonormaility ⟨vi |vj⟩=𝛿ij to the Dirac delta function:
⟨x’ |x⟩ = 𝛿(x-x’)

Question 14

Dirac Delta Function

Answer 14

-the Dirac delta function is defined as the limit
𝛿(x) = lim 𝛿ε(x)
-where ε is a subscript and the limit is taken as x tends to ε
𝛿ε(x) = {1/ε , -ε/2

Question 15

Properties of the Dirac Delta Function

Answer 15

-when we integrate any other function f, Dirac delta gives the value of f at a given point:
∫ dx 𝛿(x) f(x) = f(0)
∫ dx 𝛿(x-a) f(x) = f(a)
-and its representation in terms of plane waves:
𝛿(x) = 1/2π *∫ dk e^(ikx)
-where the integrals are taken from -∞ to +∞

Question 16

Recovering Schrodinger’s Wave Equation from the Basis Representation

Answer 16

-to describe a particle on the x-axis we must use basis |x⟩
-the state of the particle is given by ket |ψ⟩ if we represent it in our basis we get column vector with entries ⟨xi |ψ⟩
-the Schrodinger wave function is just a representation of ket |ψ⟩ in the basis |x⟩
-the probability density, the probability of finding a particle at the point x:
|⟨x |ψ⟩|²

Question 17

Operator

Definition

Answer 17

-a mathematical rule that acts on a ket vector and transforms it into another ket:
^A |ψ⟩ = |ψ’⟩
-the same applies to bras:
⟨ψ| ^A = ⟨ψ’|

Question 18

Linear Operator

Definition

Answer 18

-if the following holds for any vectors then the operator ^A is linear:
^A ( a |ψ⟩ + b |φ⟩) = a ^A |ψ⟩ + b ^A |φ⟩

Question 19

Expectation Value / Mean Value

Definition

Answer 19

-the mean value of an operator ^A in the state |ψ⟩ is defined:
⟨^A⟩ = ⟨ψ| ^A |ψ⟩ / ⟨ψ|ψ⟩
-the division by ⟨ψ|ψ⟩ is just to normalise so if ψ(x) is already normalised, ⟨ψ|ψ⟩=1 and ⟨^A⟩ = ⟨ψ| ^A |ψ⟩
-to evaluate ⟨ψ| ^A |ψ⟩, evaluate ^A |ψ⟩ = ⟨^A⟩ = |ψ’⟩, and then compute ⟨^A⟩ = ⟨ψ|ψ’⟩

Question 20

Operators in Dirac Notation

Answer 20

-objects of the form |ψ⟩ ⟨φ| are operators in Dirac notation
I..e ^A = |ψ⟩ ⟨φ|
-then ^A |ψ’⟩ = |ψ⟩ ⟨φ|ψ’⟩, and ⟨φ|ψ’⟩ is just a complex number C, so ^A |ψ’⟩ = C |ψ⟩

Question 21

Identity Operator

Answer 21

-maps every vector onto itself:
^I |ψ⟩ = |ψ⟩
-can be represented in terms of all vectors that form an orthonormal basis { |vi⟩ } of the Hilbert space:
^I = Σ |vi⟩ ⟨vi|
-where the sum is taken from i=1 to i=N and N is the dimension of the Hilbert space

Question 22

Commutator

Answer 22

-the commutator of operators ^A and ^B is:
[^A , ^B]
= ^A ^B - ^B ^A

Question 23

Matrix Representation of an Operator

Answer 23

-the matrix representation of ^A is an NxN matrix where N is the dimension of the Hilbert space
-with entries Aij such that:
Aij = ⟨vi| ^A |vj⟩
-where vi, vj are elements in an orthonormal basis of the Hilbert space

Question 24

Hermitian Adjoint

Definition

Answer 24

-a Hermitian adjoint of an operator ^A is defined as:
⟨ψ| ^A† |φ⟩ = ( ⟨φ| ^A |ψ⟩ )*
-where ^A† indicates the Hermitian adjoint and * indicates taking the complex conjugate

Question 25

Hermitian Operator

Definition

Answer 25

-a special case of operator where:

^A = ^A†

Question 26

Rules for Calculating Hermitian Adjoints

Answer 26

1) constant c become complex conjugates c*
2) ket vectors become bra vectors and vice versa
3) operators ^A become ^A†

Question 27

Operator Eigenvalues

Answer 27

-in general an operator acts on a vector and maps it to a different vector:
^A |ψ⟩ = ~ |φ⟩ ≠ |ψ⟩
-but sometimes it happens that the operator maps the vector onto itself:
^A |ψ⟩ ~ |ψ⟩
-this is the case of significance

Question 28

Eigenvectors and Eigenvalues

Operator Definition

Answer 28

-a vector |ψ⟩ is an eigenvector of the operator ^A, if:
^A |ψ⟩ = a |ψ⟩ , aϵC
a is an eigenvalue of ^A
-the equation above defines the eigenvalue problem of ^A, its solutions are eigenvalues and eigenvectors of ^A

Question 29

Spectral Decomposition Theorem

Answer 29

-eigenvectors of Hermitian operators are always real numbers
-the corresponding eigenvectors are orthogonal
-if eigenvalues are λi and corresponding eigenvectors are |λi⟩, the operator can be written as a spectral decomposition:
^A = Σ λi |λi⟩ ⟨λi|
-where the sum is taken from i=1 to i=N

Question 30

Computing Functions of Operators

Answer 30

-to compute a function of an operator ^A, (e.g. e^(^A) or √^A) first perform spectral decomposition:
^A = Σ λi |λi⟩ ⟨λi|
-then:
f(^A) = Σ f(λi) |λi⟩ ⟨λi|
-i.e. the function of an operator amounts to evaluating the function on its eigenvalues which are complex numbers

Question 31

Postulate 2

Answer 31

Observables

• observables are physically measurable quantities (energy, momentum, coordinate, etc.)
• in quantum mechanics, observables are represented by Hermitian operators acting on the Hilbert space

Question 32

Postulate 3

Answer 32

Projection
-assume the system is in the state |ψ⟩ and we measure an observable A
-the only possible results of this measurement are the eigenvalues of A that appear in spectral decomposition:
^A = Σ λi |λi⟩ ⟨λi|
-after the measurement the state of the system is projected to an eigenvector corresponding to the measured eigenvalue
-i.e. measuring the eigenvalue λn sends the system from state |ψ⟩ to |λn⟩

Question 33

Observables as Hermitian Operators

Answer 33

-the eigenvalues of Hermitian operators are always real so any experiment will only measure real numbers which agrees with intuition

Question 34

Postulate 3 and Heisenberg’s Principle

Answer 34

-according to postulate 3, measurement of the system changes its state which agrees with Heisenberg’s principle

Question 35

Postulate 4

Answer 35

Probabilistic Outcomes of Measurement
-we know that the only possible results of measuring some observable A are its eigenvalues that appear in the spectral decomposition
-if the system is in state |ψ⟩, each eigenvalue λn will be measured with probability:
P(λn) = | ⟨λn|ψ⟩ |² /⟨ψ|ψ⟩
-the average or expectation value of ^A in the state |ψ⟩ is:
⟨^A⟩ = ⟨ψ|^A|ψ⟩ \ ⟨ψ|ψ⟩

Question 36

Eigenvalue Probabilities

Answer 36

-since the only possible measured values are the eigenvalues, the sum of the probabilities of each eigenvalue must add up to one:
Σ P(λn) = 1

Question 37

When is the probability of measuring a given eigenvalue high?

Answer 37

-when the state of the system before measurement is close to the eigenstate that corresponds to the particular eigenvalue

Question 38

Coordinate Representation

Answer 38

-if the Hilbert space is a 1D line, we have an infinite number of points x which can each be associated with a ket vector |x⟩
-each point x can be thought of as the Eigen value of an operator:
^x |x⟩ = x |x⟩
***

Question 39

Momentum Operator

Answer 39

-momentum becomes a derivative operator:
^px = -iℏ ∂/∂x
-similarly for each coordinate in 3D space:
^py = -iℏ ∂/∂y
^pz = -iℏ ∂/∂z

Question 40

Uncertainty Principle From the Momentum Operator

Answer 40

[^x , ^p] = iћ * I

• where I is the identity operator
• to derive this, multiply both sides by a general wave function ψ(x) and expand and simplify

Question 41

Postulate 5

Answer 41

Time Evolution
-time evolution of a system is governed by the time-dependent Schrodinger equation:
iћ ∂|ψ(t)⟩/∂t = ^H |ψ(t)⟩
-this equation is valid for any Hamiltonian
-but in the special case that the Hamiltonian has no time dependence we can simplify to solve the equation:
|ψ(t)⟩ = e^(-i/ћ t^H) |ψ(0)⟩
-if the Hamiltonian does not depend on time we only need to solve the time-independent Schrodinger equation:
-ћ²/2m ∂²φ/∂x² + V(x)φ(x) = Eφ(x)
-to get the full solution just multiply with a phase;
ψ(x,t) = φ(x)e^(-i/ћ t^H)