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1.)
Why Study QM: QM is necessary for a more fundamental understanding of nature.
2.)
The Classical Limit: This is the limiting case where ℎ → 0. If properties of an observed system
are much larger than Planks constant, where h approaches zero, then the system will behave classically
rather than quantum mechanically. In our macroscopic world objects appear to be definite in measure.
For example, a baseball in an empty room remains in one spot regardless of the number of times an
observer measures its position, assuming there is no work being done on the ball. The baseball is within
the classical limit because it is a system comprised of a very large amount of particles giving it a large
mass.
The Correspondence Principle describes how QM relates to physics in the classical limit. It connects QM
to the macroscopic world by way of using a statistical interpretation. It states that the familiar
observations of classical physics are supported by the expectation values of the quantum systems. On
average the observed quantity in a quantum system will agree with classical predictions.
3.)
Ehrenfest’s Theorem: Ehrenfest’s theorem relates time changing variables quantum
mechanically. Specifically it relates the expectation value of the time derivative of an observable to the
expectation value of the commutation of the observable and the Hamiltonian. In light of the
correspondence principle, Ehrenfest’s Theorem provides a quantum mechanical description of the
dynamic variables of classical mechanics.
𝑖
〈𝐴̇〉 = 〈[𝐻, 𝐴]〉
ℏ
This relation is found by taking the derivative of the expectation value of the observable. While the
operator is assumed to be time independent the state vector it is operating on is time dependent. Since
〈𝐴〉 = 〈𝜓|𝐴|𝜓〉 the product rule, along with the time dependent Schrodinger equation, can be employed
in determining 〈𝐴̇〉.
6.)
The Generalized Uncertainty Relations: This stems from the mathematical relation between
uncertainties of observables in Hilbert space and their commutator. The Formula for the generalized
uncertainty principle provides a mathematical foundation for more profound uncertainty relations, such
as the momentum space uncertainty principle. The resulting generalized uncertainty principle is nonintuitive, and is a consequence of the statistical mathematics that QM is structured on.
𝜎𝐴2 𝜎𝐵2
2
1
≥ ( 〈[𝐴, 𝐵]〉)
2𝑖
4.)
Position-Momentum Heisenberg Uncertainty Relations: The position momentum uncertainty
principle can be arrived at using the generalized uncertainty relation. Replacing A and B with
momentum and position operators gives
1
2
𝜎𝑃2 𝜎𝑋2 ≥ (2𝑖 〈[𝑃, 𝑋]〉) .
The x space representation is used for operators X and P. The commutator is then [𝑃, 𝑋] = 𝑖ℏ. Putting
this back into the general relation gives the momentum space uncertainty relation.
ℏ 2
𝜎𝑃2 𝜎𝑋2 ≥ (2) or
ℏ
𝜎𝑃 𝜎𝑋 ≥ 2
What the generalized uncertainty relation implies is if two operators do not have a similar basis, then
both observables cannot be known with zero uncertainty. Since the position and momentum operators
are incompatible (do not commute), there exists an uncertainty between the two. For example, the
more one knows about the position of an electron the less they know about its momentum. This
inverse relation has profound implications. Completely localizing a particle would require infinite
energy. Likewise for one to know the momentum of a particle with certainty would require infinite
space.
5.)
Energy –Time Heisenberg Uncertainty Relations:
𝜎𝐻 𝜎𝑇 ≥
ℏ
2
This relation is found by using Ehrenfest Theorem and the generalized uncertainty relation. It appears
intuitive from the Position Momentum Heisenberg Uncertainty Principle that the energy time derivation
would follow. This is primarily because of the correspondence between space and time, and
momentum and energy. Nonetheless, time is not the same as Energy, momentum, position, and all
other dynamic variables. The uncertainty in time needs to be properly defined, and can be done so with
the following uncertainty relation. Assuming time independent operators:
2
1
ℏ 𝑑〈𝐵〉
𝜎𝐻2 𝜎𝐵2 ≥ ( 〈[𝐻, 𝐵]〉) = (
)
2𝑖
2 𝑑𝑡
𝜎𝐻 𝜎𝐵 ≥
2
ℏ 𝑑〈𝐵〉
|
|
2 𝑑𝑡
This provides an uncertainty relation between the Hamiltonian and an arbitrary operator, B, in terms of
the time evolution of the expectation value of B. By rearranging this relation, a clear definition of time
uncertainty can be ascertained.
𝜎𝐻 𝜎𝐵 |
𝑑𝑡
ℏ
|≥
𝑑〈𝐵〉
2
Consequently, time uncertainty can be described as the amount of time for the expectation value of an
observable to change by one standard deviation.
7.)
Angular Momentum Uncertainty Relations: These are also found using the generalized
uncertainty relation. The three angular momentum commutators are as follows.
[𝐿𝑧 , 𝐿𝑋 ] = 𝑖ℏ𝐿𝑦
[𝐿𝑥 , 𝐿𝑦 ] = 𝑖ℏ𝐿𝑧
[𝐿𝑧 , 𝐿𝑦 ] = 𝑖ℏ𝐿𝑥
The uncertainty relations for angular momentum are then:
𝜎𝐿𝑧 𝜎𝐿𝑥 ≥
ℏ
〈𝐿 〉
2 𝑦
𝜎𝐿𝑥 𝜎𝐿𝑦 ≥
ℏ
〈𝐿 〉
2 𝑧
𝜎𝐿𝑧 𝜎𝐿𝑦 ≥
ℏ
〈𝐿 〉
2 𝑥
𝐿𝑥 𝐿𝑦 𝐿𝑧 are operators observing the components of angular momentum. The uncertainty in
simultaneously knowing any two angular momentum component is, at best, proportional to the
expectation value of the third angular momentum component. Consequently, the question of whether
the component operators are compatible is strictly dependent on the frame of reference. For example,
if ones’ frame of reference is such that 𝐿𝑦 is zero then the z and x angular momentum components
become compatible
8.)
Boundary Conditions for Finite and Infinite Potentials:
A
C
B
D
E
F
X=0
X=a
Case 1: Finite Potential
A
B
D
X=0
E
X=a
Case 2: Infinite Potential
In order for the wave function to remain continuous in cases 1,𝜓𝐴 (0) = 𝜓𝐷 (0) = 𝜓𝐶 (0) and 𝜓𝐵 (𝑎) =
𝜓𝐸 (𝑎) = 𝜓𝐹 (𝑎). Also
𝑑𝜓𝐴 (0)
𝑑𝑥
=
𝑑𝜓𝐷 (0)
𝑑𝑥
=
𝑑𝜓𝐶 (0)
𝑑𝑥
and
𝑑𝜓𝐵 (𝑎)
𝑑𝑥
=
𝑑𝜓𝐸 (𝑎)
𝑑𝑥
=
𝑑𝜓𝐹 (𝑎)
.
𝑑𝑥
This notation is used to
point out that in a finite well there are incident, reflected, and transmitted components to the wave
function. In short 𝜓(0) = 𝑔, 𝜓(𝑎) = ℎ, and 𝜓(±∞) = 0. The last condition is necessary in order for
the wave function to be normalized. In case 2, the boundary conditions are 𝜓(0) = 0 and 𝜓(𝑎) = 0.
9.)
Recursion Relation: This relates to patter finding. Specifically, when a differential equation has
has an infinite set of solutions each with unique coefficients, a general relation can be developed that
relates a particular coefficient to coefficients of earlier terms.
10.)
Why Study the Harmonic Oscillator: The Harmonic Oscillator is import because it is prevalent in
many different areas of physics. One such area is in the study of the properties of different types of
materials. The harmonic oscillator model is useful in describing systems with complex crystalline
lattices. In addition the harmonic oscillator model gives allows for a quantum mechanical description of
covalent compounds.
11.)
The zero point energy and the zero point motion of the harmonic oscillator: The quantum
1
harmonic oscillator has discrete energy levels represented as (𝑛 + 2)ℏ𝑤, where n is any positive integer.
The Lowest energy level is called the zero point energy level and is
ℏ𝑤
.
2
This is the lowest energy the
system described by the harmonic oscillator can posses. Interestingly this value is not zero, therefore,
even at its lowest energy state, there still exists some oscillation in the system.
12.)
Factoring the Hamiltonian for the Harmonic Oscillator: This technique is used to attain the
raising and lowering operators for the harmonic oscillator. In general, the Hamiltonian is H=T+V. The
Hamiltonian for the harmonic oscillator is:
𝐻=
𝑃2 𝑚𝜔2 2
+
𝑋
2𝑚
2
Factoring this gives the raising and lowering operators:
+
𝑎−
=
1
√2𝑚ℏ𝜔
(−
+𝑖𝑃 + 𝑚𝑤𝑋)
The energy Eigenvalues can subsequently be found by representing the Hamiltonian in terms of the
ladder operators.
13-14.)
The Ladder Operators for the SHO in Hilbert and Position Space: The ladder operators
𝑑
in Hilbert space is shown above. The Latter operator in position space calls for the relation 𝑃 = −𝑖ℎ 𝑑𝑥.
+
The ladder operators in position space are then, 𝑎−
=
1
𝑑
( −ℎ
√2𝑚ℏ𝜔 + 𝑑𝑥
+ 𝑚𝑤𝑋) .
15.)
Obtaining the Eigenenergies of the SHO Using Ladder Operators: This is done by representing
the Hamiltonian in terms of the ladder operators and solving
〈𝐸0 |𝐻|𝐸0 〉 = 𝐸0 ⟨𝐸0 |𝐸0 ⟩ = 𝐸0
The Hamiltonian in terms of the ladder operators is
1
+ −
𝐻 = ℏ𝜔 (𝑎−
𝑎+ ± )
2
Using this expression to solve for the eigenvalue-eigenvector equation gives
1
−
〈𝐸0 |𝐻|𝐸0 〉 = 〈𝐸0 |ℏ𝜔 (𝑎−+ 𝑎+
± )| 𝐸0 〉
2
1
+ − |𝐸 〉
= [〈𝐸0 |𝑎−
𝑎+ 0 + 〈𝐸0 | | 𝐸0 〉] ℏ𝜔
2
Knowing that 𝑎− |𝐸0 〉 = 0, one can attain the eigenvalue for the lowest eigenstate:
ℏ𝜔
= 𝐸0
2
Next, the raising operator is used to determine the intervals between eigenenergies.
1
𝐻𝑎+ |𝐸0 〉 = ℏ𝜔𝑎+ (𝑎+ 𝑎− + ) |𝐸0 〉
2
1
= ℏ𝜔𝑎+ (𝑎+ 𝑎− + 1 + ) |𝐸0 〉
2
The above relation is arrived at by taking advantage of the fact that the commutation of the raising and
lowering operators is 1.
= (𝐻 + ℏ𝜔)𝑎+ |𝐸0 〉
1
The interval between eigenenergies is ℏ𝜔, and the values of the eigenenergies are (𝑛 + 2)ℏ𝑤.
16.)
17-20.)
Obtaining the Eigenenergies of the SHO Using via Seperation of Variables:
See attached graphs.
21.)
The minimum uncertainty state for the harmonic oscillator: Minimal uncertainty pertains to
the condition
𝜎𝐴2 𝜎𝐵2
2
1
= ( 〈[𝐴, 𝐵]〉)
2𝑖
This state is the ground state or n=0 state of the SHO. The ground state is such because the wave
function takes on a Gaussian distribution, which is a function of minimal uncertainty. Let the variance of
the X and P operators be defined as: 𝑓 ≡ (𝑃 − 〈𝑃〉)𝜓 and 𝑔 ≡ (𝑋 − 〈𝑋〉)𝜓. Also let 𝜎𝑃2 𝜎𝑋2 = ⟨𝑓|𝑓⟩⟨𝑔|𝑔⟩.
By the Schwartz inequality ⟨𝑓|𝑓⟩⟨𝑔|𝑔⟩ ≥ |⟨𝑓|𝑔⟩|2 . To attain minimum uncertainty ⟨𝑓|𝑓⟩⟨𝑔|𝑔⟩ =
|⟨𝑓|𝑔⟩|2 and this is true if 𝑔 = 𝑖𝑐𝑓. Therefore,
ℏ 𝑑
(
− 〈𝑃〉) = 𝑖𝑐(𝑥 − 〈𝑥〉)𝜓
𝑖 𝑑𝑥
Which has a Gaussian solution
𝜓(𝑥) = 𝐴𝑒 −𝑐
(𝑥−〈𝑥〉)2 𝑖〈𝑃〉𝑥
2
𝑒 ℏ
22.)
Charles Hermite, The Hermite Equation, and the Hermite Polynomials. Charles Hermite(18221901 ) was a French mathematician who was responsible for the Hermit Equation. The Hermite
equation is one of many special ordinary differential equations. Its specialty is rooted in its ability to
produce solutions that describe the SHO. These solutions are the Hermite Polynomials. The Hermite
equation, in a form catering to the SHO, is
𝑑2
𝑑
𝜆
𝐻𝑛 (𝜉) − 2𝜉 𝐻𝑛 (𝜉) + ( − 1) 𝐻𝑛 (𝜉) = 0
2
𝑑𝜉
𝑑𝜉
𝛼
where =
2𝑚𝐸
ℏ2
𝑚𝑘
,𝛼 = √ ℏ2 , and 𝜉 = √𝛼𝑥.
23.)
Why Study Angular Momentum: Angular momentum is necessary for accurately describing
spherically symmetric quantum systems. The Hammiltonian of a system is represented by the total
energy of that system. Therefore one cannot ignore angular momentum because it is an energy storage
mechanism that affects the Hamiltonian. Mathematically, angular momentum is a bit different from the
position and momentum operators in that it is a vector of operators.
24.)
Angular Momentum Ladder Operator in Hilbert Space:
𝐿+ = 𝐿𝑥 + 𝑖𝐿𝑦
𝐿− = 𝐿𝑥 − 𝑖𝐿𝑦
The ladder operators are found using the same factoring techniques used to find the ladder operators
for the SHO. To get the ladder operators for angular momentum one does a pseudo factor of 𝐿 − 𝐿𝑧 =
𝐿𝑥 + 𝐿𝑦 . This is similar to factoring the Hamiltonian for the SHO, where the Hamiltonian in this case is
described by the total angular momentum and z component angular momentum observables.
25.)
Angular Momentum Ladder Operator in Position Space:
𝐿𝑥 = 𝑌𝑃𝑧 − 𝑍𝑃𝑦
𝐿𝑦 = 𝑍𝑃𝑥 − 𝑋𝑃𝑧
Represented in position space:
𝐿𝑥 = −𝑖ℏ𝑦
𝜕
𝜕
+ 𝑖ℏ𝑧
𝜕𝑧
𝜕𝑦
𝐿𝑦 = −𝑖ℏ𝑧
𝜕
𝜕
+ 𝑖ℏ𝑥
𝜕𝑥
𝜕𝑧
Then the ladder operators represented in position space are:
𝐿+ = −ℏ
𝐿− = ℏ
𝜕
𝜕
𝜕
(𝑥 + 𝑖𝑦) + ℏ𝑧 (𝑖
+ )
𝜕𝑧
𝜕𝑦 𝜕𝑥
𝜕
𝜕
𝜕
(𝑥 − 𝑖𝑦) + ℏ𝑧 (𝑖
− )
𝜕𝑧
𝜕𝑦 𝜕𝑥
26.)
The totally antisymmetric tensor and vector cross products : The total antisymetric tensor is used to
represent the cross product of two vectors in a more general form.
𝑐 = 𝑎⃑ × 𝑏⃑⃑ = 𝜖𝑖𝑗𝑘 𝑎𝑗 𝑏𝑘
Where the Levi-Civita symbol is represented as
𝜖𝑖𝑗𝑘
1 𝑖𝑓 𝑖, 𝑗, 𝑘 = 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 123
= {−1 𝑖𝑓 𝑖, 𝑗, 𝑘 = 𝑎𝑛 𝑜𝑑𝑑 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 123
0 𝑖𝑓 𝑎𝑛𝑦 𝑖𝑛𝑑𝑖𝑐𝑒𝑠 𝑎𝑟𝑒 𝑟𝑒𝑝𝑒𝑎𝑡𝑒𝑑
What this means is that if any pair of indices are flipped an even number of times 𝜖𝑖𝑗𝑘 = 1. If a pair of
indices are flipped an odd number of times then 𝜖𝑖𝑗𝑘 = −1. A more abstract interpretation is that the
sign of 𝜖𝑖𝑗𝑘 depends on the direction of rotation on cyclic ordering of the indices. The antisymmetric
tensor relates to physical that display its symmetry such as angular momentum. For example, the sign
of the angular momentum vector in a given system is dependent on the direction of rotation, or cyclic
ordering. Therefore angular momentum is cyclic. Furthermore angular momentum is the cross product
of a position vector and linear momentum vector. This can be extended to the commutation relations
regarding the momentum operators, which can be written in terms of the Levi-Civita symbol.
[𝐿𝑖 , 𝐿𝑗 ] = 𝑖ℏ𝜖𝑖𝑗𝑘 𝐿𝑘
27.)
Angular momentum commutation relations in vector and component form:
[𝐿𝑥 , 𝐿𝑦 ] = 𝑖ℏ𝐿𝑧
[𝐿𝑧 , 𝐿𝑥 ] = 𝑖ℏ𝐿𝑦
[𝐿𝑦 , 𝐿𝑧 ] = 𝑖ℏ𝐿𝑥
28-29.) The 𝑳𝟐 and 𝑳𝒛 operators and theireigenkets/eigenfunctions and eigenvalues in Hilbert and
position space: The 𝐿2 and 𝐿𝑧 operators are two of the three operators used in describing the energy of
a system with a spherically symmetric coulomb potential like the hydrogen atom. They represent the
total angular momentum squared and its z projection. The reason why the square of the total angular
momentum is used is so that the operator is represented in the same 3 space as the component angular
momentum operators. To find the eigenkets and eigenvalues of the two operators in Hilbert space,
similar processes used with the SHO are employed. The eigenkets are represented as |𝛼, 𝛽〉 where the
index 𝛼 represents the all 𝐿2 eigenvalues and 𝛽 all 𝐿𝑧 eigenvalues. So in Hilbert space:
𝐿2 |𝑙, 𝑚〉 = ℏ2 𝑙(𝑙 + 1)|𝑙, 𝑚〉
𝐿𝑧 |𝑙, 𝑚〉 = 𝑚ℏ|𝑙, 𝑚〉, −𝑙 ≤ 𝑚 ≤ 𝑙
Regarding position space, the 𝐿2 and 𝐿𝑧 operators are represented in spherical coordinates as,
𝐿𝑧 = −𝑖ℏ
𝜕
𝜕𝜙
𝜕2
1 𝜕
1 𝜕2
𝐿 = −ℏ ( 2 +
+
)
𝜕𝜃
𝑡𝑎𝑛𝜃 𝜕𝜃 𝑠𝑖𝑛2 𝜃 𝜕𝜙 2
2
2
The eigenfunction of angular momentum are the spherical harmonics
𝑌𝑙,𝑚 (𝜃, 𝜙) = 𝐴(𝑠𝑖𝑛𝑙 𝜃)𝑒 𝑖𝑚𝜙
30-31.)
Obtaining the eigenvalues, eigenkets, and eigenfunctions using the ladder operators
and separation of variables: This is done by implementing the ladder operators in the eigval./eigvect.
equation. Due to the fact that angular momentum components do not commute, both 𝐿2 and 𝐿𝑧
operators are need to adequately observe x, y, and z angular momentum components. The 𝐿2 and 𝐿𝑧
operators do commute and therefore share a common basis, however they aren’t guaranteed to have
indistinguishable eigenvalues. Therefore, the angular momentum eigenket possesses two indices as
|𝛼, 𝛽〉. Using the latter operators, the eigenval/eigenvec equation for 𝐿𝑧 is written as
𝐿𝑧 𝐿+ |𝛼, 𝛽〉 = (𝐿+ 𝐿𝑧 + ℏ𝐿+ )|𝛼, 𝛽〉
The above is arrived at using the commutation relation 𝐿𝑧 𝐿+ − 𝐿+ 𝐿𝑧 = ℏ𝐿+ . It can be simplified further
as
𝐿𝑧 𝐿+ |𝛼, 𝛽〉 = (𝛽 + ℏ)𝐿+ |𝛼, 𝛽〉
Noting that 𝐿𝑧 |𝛼, 𝛽〉 = 𝛽|𝛼, 𝛽〉. What the this shows is how the ladder operator changes the eigenvalue
of the 𝐿𝑧 basis. The eigenvalues increment and decrement by ℏ. The same technique is used to find
how the eigenvalues of 𝐿2 change. However, the results will be different since the 𝐿2 operator
commutes with the 𝐿+ operator.
𝐿2 𝐿+ |𝛼, 𝛽〉 = 𝛼𝐿+ |𝛼, 𝛽〉
The eigenvalues of 𝐿2 do not change when the eigenket is operated on by the ladder operators. At this
point all one can deduce is the interval in which the eigenvalues for𝐿2 and 𝐿𝑧 change. Still
undetermined are the exact values of the 𝛼 and 𝛽. This is accomplished with continued usage of the
ladder operators with the assumption that any further increment or decrement of a maximum or
minimum eigenstate is zero. Therefore,
𝐿− 𝐿+ |𝛼, 𝛽𝑚𝑎𝑥 〉 = 0
2
(𝐿2 − 𝐿2𝑧 − ℏ𝐿𝑧 )|𝛼, 𝛽𝑚𝑎𝑥 〉 = (𝛼 − 𝛽𝑚𝑎𝑥
− ℏ𝛽𝑚𝑎𝑥 )|𝛼, 𝛽𝑚𝑎𝑥 〉 = 0
𝛼 = 𝛽𝑚𝑎𝑥 (𝛽max + ℏ)
In the case of 𝛽𝑚𝑖𝑛 :
𝛼 = 𝛽𝑚𝑎𝑥 (𝛽max − ℏ)
The fact that 𝛽 has maximum and minimum values is proven by the following:
⟨𝛼, 𝛽|𝐿2 − 𝐿𝑧 |𝛼, 𝛽⟩ = (𝛼 − 𝛽 2 )⟨𝛼, 𝛽|𝛼, 𝛽⟩
Where 𝐿2 − 𝐿𝑧 = 𝐿2𝑥 + 𝐿2𝑦 . 𝐿2𝑥 and 𝐿2𝑦 are the square of Hermitian operators and must have real positive
eigenvalues. Therefore (𝛼 − 𝛽 2 ) ≥ 0 or 𝛼 ≥ 𝛽 2.
By finding 𝛽max all possible eigenvalues can be known. Instead of mathematically finding 𝛽𝑚𝑎𝑥 one can
take a more comprehensive approach. 𝛽 represents the eigenvalue for the z component angular
momentum operator. Furthermore the z component angular momentum operator is anti symmetric,
evident both intuitively and mathematically through its representation by the totally antisymmetric
tensor. Therefore it makes sense that 𝛽max = −𝛽min . With that established and knowing that the
interval of 𝛽 is ℏ,
𝛽max =
𝑛ℏ
2
𝑛
Letting 𝑙 = 2 , the eigenvalue for the 𝐿2 operator becomes ℏ2 𝑙(𝑙 + 1).
To find the eigenvalues and eigenfunctions for angular momentum in positions space, separation of
variables is used. First, state vector must be transformed from Hilbert space to a function in position
space. Due to the spherical symmetry inherent in angular momentum, spherical coordinates are used.
〈𝜃, 𝜙|𝛼, 𝛽〉 = 𝑌𝑙,𝑚 (𝜃, 𝜙)
The goal is to set up a partial differential equation solvable through separation of variables. This is done
using the same operators as before, however, because the state vector is now in position space, the
operators need to be represented similarly. First, the state function is assumed to be representable as a
product of two functions. The state function is then operated on by 𝐿𝑧 , where 𝐿𝑧 is in spherical
coordinates.
𝐿𝑧 𝑌𝑙,𝑚 (𝜃, 𝜙) = −𝑖ℏ
𝜕
[𝑓 (𝜃)𝑔𝑙,𝑚 (𝜙)] = 𝑚ℏ[𝑓𝑙,𝑚 (𝜃)𝑔𝑙,𝑚 (𝜙)]
𝜕𝜙 𝑙,𝑚
Solving this differential equation will give the solution for𝑔𝑙,𝑚 (𝜙) .
−𝑖
𝜕
𝑔 (𝜙) = 𝑚𝑔𝑙,𝑚 (𝜙)
𝜕𝜙 𝑙,𝑚
𝑔𝑚 (𝜙) = 𝑒 𝑖𝑚𝜙
The second function is solved by using the ladder operator on a maximum eigenstate.
𝐿+ |𝑙, 𝑙〉 = ℏ𝑒 𝑖𝜙 (
𝜕
𝜕
+ 𝑖𝑐𝑜𝑡𝜃 ) 𝑓𝑙,𝑙 (𝜃)𝑒 𝑖𝑙𝜙 = 0
𝜕𝜃
𝜕𝜙
𝑓𝑙,𝑙 (𝜃) = 𝐴𝑠𝑖𝑛𝑙 𝜃
The eigenfunction for 𝑚 = 𝑙 is
𝑌𝑙,𝑚 (𝜃, 𝜙) = 𝐴𝑠𝑖𝑛𝑙 𝜃𝑒 𝑖𝑚𝜙
This is the eigenfucntion corresponding to the maximum m eigenvalue. Therefore subsequent
eigenfunctions can easily be found by operating with the lowering operator in position space.
32.)
Sketch and Explain the Semi-classical Vector Model for Angular Momentum for l=4. The total
angular momentum of any system can be represented by a vector. The length of this vector is the
square root of the eigenvalue of the 𝐿2 operator, which is √ℏ2 𝑙(𝑙 + 1). This eigenvalue does not
change under rotation or reflection transformations, which makes sense because the magnitude of
angular momentum in a given system is a conserved quantity. However, the 𝐿𝑧 eigenvalues do change
under rotation transformation. This is because these eigenvalues represent measurable quantities of
the projection of the total angular momentum onto the z axis. The semi–classical vector model for the
l=4 case is shown bellow.
z
4ℏ
3ℏ
𝐿 = ℏ√20
2ℏ
1ℏ
0ℏ
y
x
33-34.) Adrien-Marie Legendre, the Legendre equation, and the Legendre polynomials and the
Spherical Harmonics: The legendre equation is a special case (namely m = 0) of the Associate
Legendre Equation. The solutions to the Associate Legendre Equation are a sum of important
polynomials, called the Associated Legendre polynomials. The Legendre equation is
(1 − 𝑥 2 )𝑓 ′′ (𝑥) − 2𝑥𝑓 ′ (𝑥) + 𝑙(𝑙 + 1)𝑓(𝑥) = 0
The generating function for the Legendre polynomials is
𝑃𝑙 (𝑥) =
(−1)𝑙 𝑑𝑙
(1 − 𝑥 2 )𝑙
2𝑙 𝑙! 𝑑𝑥 𝑙
Associate Legendre polynomials can be generated from Legendre polynomials by the relation
𝑃𝑙,𝑚 (𝑥) = (−1)𝑚 √(1 − 𝑥 2 )𝑚
𝑑𝑚
𝑃 (𝑥)
𝑑𝑥 𝑚 𝑙
The Associate Legendre polynomials are important because they represent the polar component of the
spherical harmonics, and the spherical harmonics are spatial eigenfunctions of angular momentum.
35.)
Why study the hydrogen atom: The hydrogen atom is relatively simple. It is the most basic
element and, therefore, the easiest to find solutions to regarding its quantum mechanical behavior.
Furthermore, due to its basic nature, it provides some intuitive insight that may prove useful in solving
more complex quantum mechanical systems.
36.)
The asymptotic decay of the spatial wavefunctions for hydrogen and for the SHO: For the
eigenfunctions of SHO and Hydrogen atom to be physical they must have a finite integral over all space.
What this means is that the wave functions of the SHO and Hydrogen atom decay to zero as r goes to
infinity. If this were not the case then the wave functions could not be normalized and could not exist in
Hilbert Space and, therefore, could not be physical.
37.)
The n, l, and m quantum numbers for the hydrogen atom : The wave function for the
hydrogen atom is only partially described by the spherical harmonics. The complete wave function
includes a radial component which is caused by the hydrogen atoms coulomb potential. The quantum
numbers are indices that specify the eigenvalues and corresponding eigenfunctions for the hydrogen
atom. l and m relate to the eigenvalues and eigenfunctions of the total and z component angular
momentum of the hydrogen atom. n and l relate to the radial eigenvalues and eigenfunctions of the
hydrogen atom. n is the quantum number pertaining to the total energy possessed by the system. The
total angular momentum can range from 0 to n while the z projections are between ±𝑙.
Another point of view is that the hydrogen atom is a system of one proton and one electron with a given
quantized potential energy of
13.6𝑒𝑉
.
𝑛2
This energy can distribute itself in specific quantities between
radial potential and the total angular momentum. As a result, and as a side note, for each energy level,
there are a number of effective potentials that are proportional to the number of allowed values of total
angular momentum. The m quantum number only specifies the orientation of the total angular
momentum.
38.)
Edmond Laguerre, the Laguerre equation, and the Laguerre polynomials: The Laguerre
equation is yet another special differential equation. It is of the form
𝑥𝐿𝑗′′ (𝑥) + (1 − 𝑥)𝐿𝑗′ (𝑥) + 𝑗𝐿𝑗 (𝑥) = 0
The equation has an infinite set of solutions called the leguerre polynomials. These polynomials can be
found by the generating function:
𝐿𝑗 (𝑥) = 𝑒 𝑥
𝑑 𝑗 −𝑥 𝑗
𝑒 𝑥
𝑑𝑥 𝑗
The associated Leguerre polynomials, 𝐿𝑗𝑘 (𝑥), are solutions to the associated Leguerre equation
′′
′
𝑥𝐿𝑗𝑘 (𝑥) + (1 − 𝑥 + 𝑘)𝐿𝑗𝑘 (𝑥) + 𝑗𝐿𝑗𝑘 (𝑥) = 0
The generating function for the associate Leguerre polynomials is
𝐿𝑗𝑘 (𝑥) = (−1)𝑥
𝑑𝑗
𝐿 (𝑥)
𝑑𝑥 𝑗 𝑗+𝑘
Associated leguerre functions are a normalized form of the associated Leguerre polynomials. What is
used to describe the radial wave function of the hydrogen atom is a slightly modified version of the
associated leguerre functions. The generating function for these is
𝑥 𝑘+1
2 𝐿𝑘 (𝑥)
𝑗
𝑦𝑗𝑘 (𝑥) = 𝑒 −2 𝑥
Where the quantum numbers n and l are related by
𝑘 = 2𝑙 + 1
𝑛 =𝑗+𝑙+1
39-40.) What bothers me about QM and my favorite topic in QM: At this point nothing bothers me
about quantum mechanics. I cannot come to any useful descriptions of how everything in the universe
are an outcome of interfering probability amplitudes. Therefore, I have just accepted the fact that,
despite the little sense it does make, that’s how nature operates. My favorite topic is the Hydrogen
atom. studying the hydrogen atom was no easy task, but I enjoyed the results quantum mechanics gave
because it shed light on the importance of symmetry to physical systems. Also the true structure of the
hydrogen atom was always very confusing to me.
Feynmen’s Lectures on QED
Chapter 1 of QED serves as an introduction to the forthcoming topics on light and quantum mechanics.
Richard Feynmen explains the goal of quantum theory as well as the goal of any theory in physics. That
is, not to explain why phenomena happen, but to try and extrapolate the mechanisms by which those
phenomena exist. He uses, as an analogy, the simple techniques employed by the Myans for predicting
celestial periods. Feynman notes that the ability to predict events in nature does not constitute a
fundamental understanding of the event. In the same way the Myans ability to predict the behavior of
the stars stemmed entirely from simple counting, quantum mechanics serves as a mathematical system
by which particular observations in nature can be further defined. Feynman establishes that QED,
although a highly accurate and developed model, can only explain how nature dose what it does, but
not why it chose to do it that way.
Feynman continues his introduction into QED with examples of optical phenomena that suggest that
light behaves as both particle and wave. He explains that light is known to behave as a particle by how it
interacts with photo detectors at low intensities. For example, at such intensities photo detectors only
pick up discrete random light signatures as opposed to a low level continuous signal. He counters this
with an experiment where light travels through multiple media. The detectors would pick up reflected
light varying in intensity with respect to the distances between reflective boundaries. This could only be
explained if the light was structured as a wave. Feynman argued that if one were to think of light as a
particle in this experiment, then the reflection coefficient, or the probability of detecting a reflected
photon, should remain constant.
In the second chapter Feynmen explains the previous example from a quantum mechanical perspective.
He begins with introducing a system by which the probability of whether the photon is transmitted or
reflected is represented by the length of a vector. The number of vectors is dependent on the number
of paths that the photon could possibly take to reach the detector. Feynmen explains that the vector
rotates with proportion to the change in time between each path. The resulting, time evolved, vectors
are then added and the sum vector is squared to get the probability of a photon being detected. This
can be done with reflection/transmission points separated at different distances and the results will
agree with experimental observation. Rather than focusing on the photon or wave and trying to
decipher the details of their mechanics from a duality perspective, Feynmen uses this abstract sum of
vectors model that describes events with probability amplitudes. What is meant by describing events is
determining the probability of something happening, in this case the detection of photons, in place of
certain conditions.
Using the sum of vectors technique, one has to consider all possible paths that light could travel from
point A to point B. This seems contradictory to observation, however this approach produces agreeable
results, which is what matters. It is shown that the vectors from the shortest paths will contribute the
most to the length of the vector sum, which describes the probability amplitude of an event. This is so
because the time differences between the shortest paths are small compared to other possible paths.
Other optical phenomena such as diffraction gratings and the focusing of light can be described by
adjusting either the length of the vectors (probability amplitudes) and/or the angle at which those
vectors have rotated. Changes in these variables translate to physical changes in the system. As a
result the behavior of light is greatly simplified to probabilistic outcome of an event
In chapter three Feynmen introduces Frame work for QED. He establishes three simple rules: A photon
goes from point A to point B in space-time with some probability, an electron does the same, and an
electron can absorb or emit photons at locations in space-time with some probability, which is
approximately 0.1. These three actions each occur with some probability. The probability relating to a
photon going from A to B is a simple function inversely proportional to the Interval. The function
describing the probability for an electron to go from A to B is not mentioned, due to its complexity, but
this function is different in that mass is included among its temporal and special variables. When mass
goes to zero the function describing the electron will be equivalent to that which describes the photon.
The reason for the difference in functions is that electrons can emit and absorb photons. Therefore the
possible paths between events are made more numerous and complex, which alters the net probability
amplitude between events A and B. The third action is described by a constant, which is the probability
amplitude for a photon to be emitted or absorbed. This constant is called the electron charge.
This framework is fundamental. It is abstract in that it describes events. Moreover, it describes the
probability of any event as the sum of the probabilities of all possible paths in space-time to that event.
At large intervals, the contributions from the erratic, nonsensical paths are canceled out, but they are
more evident at small intervals. This has some weird implications such as light being able to travel faster
and slower than c and electrons being able to go back in time. These time traveling electrons where
discovered to be positrons. Even thought the Feynmen diagrams make it appear that the positron is
traveling backward in time, we observe them traveling forward but with positive charge. For example in
order for a pair of electrons to behave the same way as an electron, positron pair, either time would
have to be reversed or the charge on one of the electrons would have to be negated. Since positrons
are observed in the forward direction of time they’re charge must be positive.
In the fourth Chapter Feynmen shows some problems with QED. He also illustrates how the framework
of QED might work with the rest of physics. The main problem with the theory resides with its ability to
explain certain anomalies with mathematical rigger. For instance, Electrons are measured to have some
mass, called their experimental mass, but this mass is not a reflection of the electrons ideal mass. The
reason for the discrepancy is attributed to having to include all possible paths to determine the
electrons probability amplitude at the measuring event. However when including the paths where the
spatial separation of coupling events is zero, the correction terms become infinite. This leads to a
theoretical calculation of the electron’s experimental mass to be infinite! Furthermore the theoretical
calculation for charge, or the probability for coupling, blow up to infinity. Renormalization provides as
patch work to the dilemma, but a more integral mathematical explanation is still needed.
Despite these blemishes, QED remains consistent in agreeing with experimental observation so the
theory remains. Feynmen goes on to explain how QED was used to explain the behavior of the nucleus
(protons and neutrons). The problem involved with understanding the proton was that experiments
were producing difficult results. For instance over 400 new particles were being created from proton
collisions. Using QED a new theory that explained protons was developed. This theory was known as
QCD or quantum chromodynamics. QCD proposed that the proton and neutron were made up of three
quarks, each with charge -1/3 or +2/3. QCD served to clean up the mess of new particles that were
being experimentally discovered by predicting what their fundamental building blocks were.
Fundamentally, QCD helped unveil the different types, colors, and flavors of quarks and neutrinos, as
well as the W, Z, and strong force.
Despite the success of QED one thing that still remains a mystery is the accompanying masses of the
fundamental particles. QED cannot explain the mechanism for each particle having the particular mass
that it has. In addition, QED in its current form cannot be used on describing gravity. As Feynmen
states, “This is a very interesting and serious problem”.