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1.)
The continuous time-evolution of the state vector in Hilbert space: time evolution effects the
phase of the basis dependent eigenvectors that describe state vector. Alternatively, the time evolution
of the state vector is characterized as oscillations in the imaginary plane. The frequency of oscillation is
dependent on the energy Eigenvalue of the corresponding Energy Eigenvectors of the state vector. The
evidence of time evolution can be seen when making measurements on a state vector in a linear
combination of an operator’s Eigenvectors. If the operator is non-degenerate the difference in the
phase frequencies will cause the state vector to oscillate between multiple Eigenvectors. This dynamic
property of the state vector caused by its time evolution makes the outcome of measuring uncertain.
2.)
The discontinuous reduction of the state vector: When a measurement is made on a particle
the state of that particle collapses into one of the Eigenvectors of the operator. It is called
discontinuous due to the fact that before making the measurement the system was in a generic state in
Hilbert space, but when making the measurement the system instantaneously takes on one of the
Eigenstates of the operator. The system can only be represented by the basis of the operator used in
measuring its state. Mathematically the system is instantaneously projected onto the basis of the
operator. Therefore the information of the system is confined to the eigenvectors of the operator; the
measured results will only be the Eigenvalues corresponding to the Eigenvectors of the operator (with a
certain probability).
3.)
The double slit experiment: This experiment illustrates the non intuitive effects of quantum
mechanics on the physical world. In particular, it stands as experimental proof of the wave particle
duality. A beam of electrons emitted towards a general area on a wall containing two slits. Behind the
wall is a collector to detect the electrons passing through the slits. Classically, it is expected that the
amplitude of particles detected would increase in magnitude towards the area directly behind each slit.
However experimental results showed that the amplitude of particles detected having oscillating
maxima and minima emanating out from the center of the detector.
Source
Source
Classical expectation
Experimental result
E
The patterns indicated that the electron underwent destructive and constructive interference
similar to
a wave traveling through the same double slits.
4.)
Measurement and collapse of the wave function in the double slit experiment: If an
experimenter were to measure the position of the particle right before the particle pass through either
slit the distribution pattern of the detector will resemble that of the classical expectation!!! According
to quantum mechanics the mere act of measuring collapses the wave function from the Hilbert space to
an Eigenstate of the position space. In other words the electron only acts like a wave, and therefore
interacts with the double slit like such, when the experimenter is not looking otherwise it will act like a
particle! This phenomena is an excellent illustration of the discontinuous reduction of the state vector
for if the reduction of the state vector weren’t continuous, one could measure the position of the
particle before its state actually took on the measured position Eigenvalue. For instance, if the
reduction of the state vector were continuous, one might measure the position and find that the particle
is not in front of a slit, but have it appear on the detector anyway.
5.)
Vector spaces and Hilbert spaces: Vector spaces are what are used in the mathematics of
quantum mechanics. Vector spaces provide discrete building block (Eigenvectors) to represent the state
of classical systems. In other words, the mathematics of quantum mechanics describes a classical
system as a vector in a certain n dimensional Vector space. The Hilbert space is an arbitrary, infinite
dimensional vector space that a classical system or a state vector lives in. Hilbert spaces have to be
infinite dimensional because the state vector is an infinitesimal system. In order to accurately describe a
continuous system in a discontinuous vector space, there must be an infinite number of dimensions, or
Eigenvectors.
6.)
dirac notation, bras, and kets: Dirac notation uses bras and kets as a higher level notation of
doing linear algebra. Bras are defined as column vectors, and are used in quantum mechanics to
represent the state vector. Bras are the adjoint of any kets. A bra multiplied by a ket is similar to
performing a scalar product. Also a bra multiplied by a ket represents a functional where a state vector
(the ket) is taken as a parameter and the result is a scalar, or function depending if the bra is a variable
itself.
For example:
⟨𝜓(𝑡 = 0)|𝜓(𝑡 = 0)⟩ = 1
⟨𝑥|𝜓(𝑡 = 0)⟩ = 𝜓(𝑥, 𝑡 = 0)
7. and part of 17)
Changing basis using Dirac notation: When measuring a state vector with two
different non commuting operators the state vector will undergo a change of the basis that describes it.
Ultimately changing the basis of and operator or a state vector is like changing our perspective.
Changing basis is seen when diagonalize operators. The basis of an operator is changed by what is
known as a unitary transformation:
𝑈 † 𝑋𝑈 = 𝑋 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑖𝑧𝑒𝑑
NOTE:
𝑈 † 𝑈 = 𝐼 and 𝑋 is Hermitian to ensure diagonalization is possible
The unitary matrix is comprised of the Eigenvectors of the operator being transformed. In such a
transformation the Eigenvalues (the measurable values) do not change. The only things affected are the
Eigenvectors, which are changed into unit vectors. This makes it possible to determine the Eigenvalues
of the operator by inspection and makes the matrix algebra much easier.
8.)
Complex conjugation and the adjoint operation: Complex conjugation is where one the
negative of all imaginary values in a matrix or scalar. The adjoint operation is where one operates on a
matrix with the complex conjugate transposed of that same matrix. The adjoint operation for a scalar is
just the complex conjugate of that scalar times the original scalar. The Adjoint operation will always
produce either a real valued matrix, if the operator is Hermitian, or a real valued scalar.
9.)
Inner products outer products and the projection operator:
Inner product: also known as scalar, or dot product. The inner product produces a scalar and is
represented by a bra multiplied by a ket (braket).
Outer product: Is a ket multiplied by a bra and produces a matrix.
1 ⋯ 0
|1〉〈1| = ( ⋮ ⋱ ⋮ )
0 ⋯ 0
Projection operator: Any n-dimensional vector space has n projection operators. A projection operator
is the outer product of any eigenvector belonging to a vector space. If the sum of all projection
operators from a given vector space is taken, then the resulting matrix is the identity matrix with
dimension equal to that of the vector space. A projection operator, made from the outer product of the
ith Eigenvector of a vector space, operating on a state vector will project the value of the ith component
of the state vector onto the projection operator.
𝑤1
|𝑤〉 = (𝑤2 ) ,
𝑤3
𝛿1𝑛
|𝑛〉 = (𝛿2𝑛 )
𝛿3𝑛
𝛿1𝑛
|𝑛〉〈𝑛| = ( 0
0
𝑤1 𝛿1𝑛
𝑃𝑛 |𝑤〉 = ( 0
0
0
𝛿2𝑛
0
0
0 ) = 𝑃𝑛
𝛿3𝑛
0
𝑤2 𝛿2𝑛
0
0
0 )
𝑤2 𝛿3𝑛
Projection operators are used to show how a state vector can be represented as a sum the Eigenstates
of any particular Hermitian operator.
10.)
Hermitian operators and physical observables: Hermitian operators are square matrices that
are equal to their adjoint. Such operators can represent a physical observable. What is meant by a
physical observable is any physical measurement that can be made on a physical system (position,
momentum, spin, etc.). All Hermitian operators have REAL eigenvalues and orthogonal bases. As a
consequence of only having real eigenvalues, and the fact that all observable quantities are real, any
observable quantity can only be described by a Hermitian operator.
11.)
A complete set of states (continuous and discrete): A complete set of discrete states pertains
to a set of linearly independent vectors that span a given vector space. In other word the set of linearly
independent vectors constitute a basis for a given vector space from which any state vector in such a
vector space can be described. Due to being linearly independent and comprising the basis of a given
vector space, a complete set of states are orthogonal. Alternatively:
𝑛
1. ) ∑⟨𝑖|𝑗⟩ = 𝛿𝑖𝑗
𝑖=1
1) is true for 1 ≤ j ≤ n where n is the dimension of the vector space.
A complete set of continuous states are a set of orthonormal functions that CONVERGE to
adequately describe a more complex function. In quantum mechanics the complex function is the
state vector in function space. Saying that the complete continuous set converges to describe the
system is analogous to saying a complete set of basis vectors describe any given vector in the vector
space they span. Likewise a complete set of continuous states comprise a basis for the function
space the continuous system resides in. Testing whether basis functions are orthogonal:
∞
2. ) ∫ 𝑓𝑖 (𝑥)𝑓𝑗∗ (𝑥)𝑑𝑥 = 𝛿𝑖𝑗
−∞
2) is a test for orthonormality and is true for all Eigenfunctions belonging to a complete set.
12.)
Resolutions of the Identity (continuous and discrete): Resolution of the identity is simple
meant to mean another way of expressing the identity operator. Resolution of the identity operator Is a
useful technique employed in doing matrix calculations pertaining to quantum mechanics. An example
is the completeness relation:
∑ 𝑃𝑖 = 𝐼
𝑖
where Pi represents the projection operator. Since the projection operator will only contain unit
vectors, the sum of all n projection operators in a given n dimensional vector space will produce the
identity matrix. Resolution of the identity is also useful for transitioning from the abstract Hilbert space,
which direct notation is very useful, to a more specific, continuous subspace, where the use of functions
dominate. An example is representing the inner product in continuous position space through the use
of resolution of the identity:
⟨𝜓|𝜓⟩ = ⟨𝜓|𝐼|𝜓⟩
∞
= 〈𝜓| (∫ |𝑥〉〈𝑥|𝑑𝑥 ) |𝜓〉
−∞
Since x is represents a continuous position space.
= ∫⟨𝜓|𝑥⟩⟨𝑥|𝜓⟩ 𝑑𝑥
Where ⟨𝑥|𝜓⟩ is a functional producing a function of x and ⟨𝜓|𝑥⟩ is the complex conjugate.
= ∫ 𝜓 ∗ (𝑥)𝜓(𝑥) 𝑑𝑥
13.)
The canonical commutation relations: an operator, specifically one arrived at from a
commutation of two other operators, which is equivalent to a multiplicative factor of ±iℏ. The position
and momentum operators are an example of having canonical commutations with each other.
Canonical commutators are the cause for the uncertainty principle and therefore the strangeness
inherent in quantum mechanics.
14.)
Degenerate Hermitian operators: A Hermitian operator said to be degenerate means that the
operator shares the same eigenvalue for different eigenvectors. With degenerate operators it is hard to
know which eigenvalue corresponds to which eigenvector. This is important because eigenvalues
represent the value obtained from making a measurement with that particular operator. With
degenerate operators one could measure a result but cannot confirm with 100 percent certainty what
the state of the system is in based off the result of that measurement. The shortcoming of degenerate
Hermitian operators is remedied through the arduous process of simultaneous diagonalization.
However this is only possible if there exists a non-degenerate Hermitian operator that the degenerate
one commutes with.
15.)
Simultaneous diagonalization of Hermitian Matrices: As mentioned above this is powerful
method in diagonalizing Hermitian matrices. The process involves finding the unitary operator of one of
the two commuting Hermitian operators, then using the unitary operators to diagonalize both matrices.
The two operators, A and B, must commute, or share the same basis, otherwise the unitary operator for
A will not diagonalize B and vice versa. A unitary operator is constructed by the eigenvalue/eigenvectors
of a Hermitian matrix.
16.)
Complete set of commuting observables: The commutation of two operators is shown as:
[𝐴, 𝐵] = (𝐴 ∗ 𝐵 − 𝐵 ∗ 𝐴)
Two operators are said to commute if[𝐴, 𝐵] = 0. Commuting operators will share the same basis. A
complete set of commuting observables is necessary when decoding degenerate matrices. Such a set
consists of a group of Hermitian matrices with common basis where at least one matrix is not
degenerate. If such a set exists then any degenerate matrix in the set can be diagonalized through the
process of simultaneous diagonalization.
17.)
Unitary operators, change of basis, and time translation: (see 7 for unitary operators and
change of basis) time translation can be viewed as a type of transformation. In quantum mechanics it
deals with the time evolution of quantum mechanical systems (state vectors) as well as operators.
Particularly, time translation will affect the outcome of measuring a state vector comprised of a linear
combination of the observable’s eigenvectors. Time translation does not affect the probability of
measuring a certain eigenstate. Moreover, the Hamiltonian of a system is invariant under time
translation thanks to our friend conservation of energy.
18.)
The Propagator: the propagator gives the probability amplitude for a particle to travel from
one place to another in a given time, or to travel with a certain energy and momentum (Wikipedia).
19.)
Compatible, partially incompatible, and completely incompatible operators: Compatible
operators share the same Eigenstates and therefore commute and can be diagonalized by the same
transformation. Incompatible operators do not share Eigenstates, thus do not commute, and thus
cannot be simultaneously diagonalized. Partially compatible share only some Eigenstates, but not all.
20.)
Position, momentum, and Energy Bases: Position, energy, and momentum are all observable.
All observable can be represented by a Hermitian matrix. The position and energy operators do not
commute and so do not share the same bases. The energy and momentum operators do share the
same basis. The commutator of the position and momentum operator are:
[𝑋, 𝑃] = 𝑖ℏ
In position space
In momentum space
𝑋=𝑥
𝑃=𝑝
𝑃 = −𝑖ℏ
𝑑
𝑑𝑥
𝑋 = 𝑖ℏ
𝑑
𝑑𝑝
21.)
Eigenvalues, eigenvector, eigenfunctions eigenkets, and eigenbras: eigenvalues are the scalars
(real for hermitian operators) values associated with each eigenstate. All observable have real
eigenvalues. The fundamental building blocks of more complicated systems are called eigenvectors
(associated with matrices), eigenfunctions (associated with continuous subspace of the Hilbert space),
eigenkets (associated with dirac notation), or eigenbras (the adjoint of its corresponding eigenket). A
slightly more formal definition of an eigenstate is given.
𝐴|𝑋〉 = 𝑎|𝑋〉
Where A is the operator where |𝑋〉 one of its eigenvectors and 𝑎 is is the corresponding eigenvalue. A
complete set of eigenstates are orthogonal to all other eigenstates.
Given a certain operator any state vector can be represented by a weighted sum of the operator’s
eigenstates.
|𝜓〉 = 𝐼|𝜓〉 = ∑|𝑎𝑖 〉〈𝑎𝑖 | 𝜓〉 = ∑|𝑎𝑖 〉 𝑐𝑖
𝑖
𝑖
23.)
Measurement possibilities and measurement probabilities: The above relation represents the
state vector in its stationary states. This is convenient because one can determine the possibilities and
probabilities by inspection. The possibilities are all possible eigenstates, |𝑎𝑖 〉, while the probabilities are
the absolute value of the factors associated with each eigenstate.
𝑃(𝑎𝑖 ) = |〈𝑎𝑖 |𝜓〉|2
Assuming the state vector is normalized.
2
= ∑|〈𝑎𝑖 |𝑎𝑗 〉𝑐𝑗 |
𝑗
Due to the face the eigenvectors are orthogonal.
= |𝑐𝑖 |2
22.)
Analogy between classical normal modes and, and quantum stationary states: Normal modes
are like the macroscopic version of stationary states. Classical normal modes can be seen in molecular
vibrations. Imagine for a moment, that a molecule represents our quantum mechanical operator. Then
each oscillatory degree of freedom for the molecule (asymmetric and symmetric flexing, stretching, etc.)
is representative of eigenvectors. This is due to the fact that the normal modes of the molecule are
completely independent of each other. Now, if some signal were to be absorbed by the molecule each
normal mode would oscillate with some magnitude depending on the spectral make up of the signal.
The molecule is describing the signals harmonics with respect to its normal modes just as an operator,
specifically the Hamiltonian, can describe a state vector in terms of its stationary states, or energy
eigenstates. In summary classical normal modes and quantum stationary states are the building blocks
of the Hamiltonian operator.
24.)
The time-independent Schrodinger equation: The TISE expresses the wave function in only one
frame of time. The wave will only vary with the variable of the subspace it is represented in. The TISE is
developed starting with the eigenvalue equation of the Hamiltonian:
𝐻Ψ = 𝐸Ψ
Where the total energy, E, of the system is the sum of the kinetic and potential energy.
𝐸 =𝑇+𝑉
𝐸Ψ = (
Represented in position space the TISE is:
𝑝2
+ 𝑉(𝑥)) Ψ
2𝑚
= (−
ℏ2 𝜕 2
+ 𝑉(𝑥)) Ψ
2𝑚 𝜕𝑥 2
25.) Solving TISE by finding the stationary states: In solving the above TISE we will inevitably find
the stationary states of the Hamiltonian in position space. If we take the potential to be zero then solve
for ψ the stationary states will be oscillatory function. If there is a potential present then the problem
becomes restricted to localizations of its subspace. For instance the oscillatory functions only exist for
E>V, while exponential terms exist for E<V. Assuming V(x) is zero we get.
𝐸Ψ = −
−
ℏ2 𝜕 2 Ψ
2𝑚 𝜕𝑥 2
2𝑚𝐸
𝜕2Ψ
Ψ
=
ℏ2
𝜕𝑥 2
So the General solution is:
𝜓(𝑥) = 𝐴𝑒
2𝑚𝐸
−𝑖√ 2 𝑥
ℏ
+ 𝐵𝑒
2𝑚𝐸
𝑖√ 2 𝑥
ℏ
From the above solution one can further develop the stationary states,𝜓(𝑥), depending on the
parameters of the problem (SHO, infinite square well, scattering).
26. and 27.)
The TDSE: TDSE incorporates the time evolution of the stationary states. It describes
the change in 𝜓 with respect to time. TDSE is shown as
𝐻Ψ(x, t) = iℏ
∂
Ψ(x, t)
∂t
⇒ 𝐸Ψ(x, t) = iℏ
∂
Ψ(x, t)
∂t
Solving this differential equations gives
Ψ(x, t) = 𝜓(𝑥, 𝑡 = 0)𝑒
−𝑖𝐸𝑡
ℏ
More generally:
Ψ(x, t) = ∑ 𝜓𝑖 (𝑥, 𝑡 = 0)𝑒
−𝑖𝐸𝑖 𝑡
ℏ
𝑖
Where 𝜓(𝑥, 𝑡 = 0) is the stationary state of the wave function. The probability amplitude of a
stationary state does not change with time since the time dependent factors cancel when squaring the
wave function. However, when the wave function is composed of a sum of multiple stationary states,
which is most likely the case, its probability amplitude will be time dependent. This is because
stationary states have different time dependent phases corresponding to different Energy eigenvalues.
These phase offsets become apparent when squaring the wavefunction thus preserving time
dependency.
28 and 29.) When a wave function is a linear combination of stationary states then the position and
energy expectation value will be time dependent. With the particle in the box experiment, the
expectation value will oscillations with in an infinite square well only when there are multiple
eigenstates present. Since the expectation values is the tether that links the quantum world to our
classical world, this time dependent, oscillating property must make macroscopic sense. In fact it does.
The oscillations of the expectation values are analogues to the particle moving back forth within the
“box”, the particle accelerating and decelerating. There is also a time dependence in the uncertainty.
This makes sense because there’s less localization in position space of the particle with higher velocities
than lower velocities. Since the expected position of the particle is in oscillation, its uncertainty in
position will oscillate. Moreover the uncertainty in the expected energy will oscillate. It will be at a
minimum when the particle is at its maximum velocity, and will be at a maximum when the particle is
localized (at its minimum velocity). Furthermore, the time dependent dynamics of the particle’s position
and energy uncertainties illustrate the canonical relation of the two subspaces.
30-33.)
The Postulates of quantum mechanics:
First postulate:
A particle in quantum mechanics is represented mathematically as a vector in an arbitrary infinite
dimension space called a Hilbert space. Like the classical particle, which from it can be measured all
possible observables; the Hilbert space is an arbitrary space containing the subspace, or infinite
dimensional vector space, of all observables.
Second postulate:
Anything you can measure classically is represented in the mathematics of quantum mechanics as a
Hermitian operator, or matrix. Observables are Hermitian operators because such matrices produce
real eigenvalues, which is what actually gets measured.
Third postulate:
Operators are like a bag of cookie cutters and a state vector is like a wad of cookie dough. Operating on
the cookie dough with bag A will only produce shapes contained in bag A.
Fourth postulate:
In operator bag A there are x amount of cookie cutters 1, y amount of 2, and z amount of 3. With each
cut you pick at random a cookie cutter. Therefore there is a probability associated with cutting shape 1,
2, and 3. In quantum mechanics this probability is dependent on the absolute value of the inner product
of the state vector and the corresponding eigenvector squared.
Fith postulate:
If after a cut the cookie dough takes on the shape of cutter 1, 2, or 3 then the dough is now in the shape,
or state, of cutter 1, 2, or 3!
Sixth postulate:
This is where my creativity comes to an end. Sixth postulate deals with the time dependent Schrodinger
equation:
𝐻|Ψ〉 = 𝑖ℏ
𝑑
|Ψ〉
𝑑𝑡
From this formula we can find out everything there is to know about a particle, and represent that
information as a function in our subspace of choosing.