Quantum physics Flashcards
What can a spatial quantum state be described by?
A wavefunction
Are all quantum states vectors?
Yes
What are the rules for a vector space?
The addition rule (the sum of two vectors in the set is also in the set), and scalar multiplication (addition and multiplication rules come with sub-properties)
How can we see a wavefunction as a vector?
The wavefunction produces a number for every value of the position x, and these numbers could all be put into an infinite column vector
What is the name and notation of the Dirac notation that is used to write a vector in a possibly infinite- dimensional, possibly complex vector space?
A ket and it is written like |u>
Can kets be used to represent a particular finite dimensional vector or an infinitely large vector or both?
Both
Is the ket notation basis-independent or basis-dependent?
Independent
How do you find the inner product for finite dimensional complex vectors?
Find the adjoint (complex conjugate of the transpose of the vector) of the first vector and multiply it by the second
If |u> and |v> are orthogonal, what does their inner product equal?
Zero
How do you calculate the inner product for infinite-dimensional vectors |u> and |v>, corresponding to wavefunctions u(x) and v(x)?
The integral between infinity and minus infinity of the complex conjugate of u(x) multiplied by v(x)
To ensure that the inner product exists for all pairs of vectors (so it doesn’t diverge), we can require that the norm of all vectors is what?
Finite
What are square-integrable functions?
The integral of the absolute value of the vector squared is finite
What is the vector space for wavefunctions restricted by?
Square-integrable functions
To represent a physical system, a normalised vector is used, which means the norm is always equal to what?
1
The vectors in a Hilbert space are what?
Quantum states
A vector space with an inner product, and which is complete, is known as what?
A Hilbert space
What is the bra?
The Hermitian conjugate, or adjoint, of the ket, |u>
A bra next to a ket gives what?
The inner product
What are the space of bras called and what are the elements that live in it called?
Dual vector space and dual vectors
The inner product in linear in its second argument, which means the the bra represents what?
A linear function
The adjoint of a bra is what?
A ket
The adjoint of a ket is what?
A bra
Are bras basis dependent or independent?
Independent
What is the difference between a bra and a ket?
A bra is a dual vector, or a function on vector space, while the ket is an actual vector on which the function can act
What is a basis?
It is a linearly independent spanning set, so any vector in a vector space can be written uniquely as a linear combination of basis vectors
What does the number of vectors in a basis equal?
The dimension of the vector space (for example, the Hilbert space)
What is the orthonormal basis set?
The vectors in the basis are orthogonal and normalised, so the inner product of any two basis vectors is equal to the Kronecker delta function
When we talk about a basis for Hilbert space, what can we assume about it (in this course)?
That it is orthonormal
How many orthonormal bases exist for every Hilbert space?
Infinity
If you decompose some general vector, v, in an orthonormal basis, how do you calculate the scalar coefficient of a particular basis vector?
Calculate the inner product between the particular basis vector and the general vector, v
For an infinite-dimensional space, if the basis states can be labelled by integers, what is the number of basis states said to be?
Countably infinite
For an infinite-dimensional space where the basis states are labelled by real values, rather than integers, the number of basis states is said to be what?
Uncountably infinite
For uncountably infinite basis states, what do we have to change?
The Dirac delta function (rather than the Kronecker delta function to show orthonormality) and replace any sum with an integral
What is the coefficients of the decomposition of the wavefunction state in the position state?
The wavefunction
How can we show the decomposition of the wavefunction state in the position basis?
The integral of the position basis multiplied by the inner product of the position basis and the wavefunction dx
What is the scalar factor that is common when switching between the position and momentum basis for the wavefunction?
1 over root(2 x pi x h bar)
What does an operator do in Hilbert space?
It acts on a vector and gives another vector
What symbol is used in Dirac notation to show that an object is an operator?
A hat above the letter signifying it
What is an example of an operator?
Matrices
Do matrices act linearly on vectors?
Yes
In this course, when we use the term ‘operator’, what can we assume about it?
That it is linear
What is the identity operator?
It takes any vector into itself
What is an example of the identity operator using vectors in a basis?
The sum over all basis vectors of the basis vectors multiplied by the corresponding dual vector
Are operators basis-independent or dependent?
Independent
What is it called when you write the identity operator in terms of different basis sets?
Using different resolutions of the identity
How can you write decompositions using the identity operator?
The vector is equal to the identity operator multiplied by the vector
How can operators be defined?
By their actions on a basis set
How do you find the matrix elements of an operator?
The ijth element is made from the i dual vector multiplied by the operator then multiplied by the j vector, which altogether make a matrix in the j vector basis
In terms of the matrix elements, the Hermitian conjugate of an operator equals what?
The complex conjugate of its transpose
What is the Hermitian conjugate of a scalar?
Its complex conjugate
How is the Hermitian conjugate of a operator written?
With a dagger
How do you find the Hermitian conjugate of any sequence of bras, ket and/or operators?
Reverse the order of the components then take the Hermitian conjugate of each
What are observables in quantum theory represented by?
Hermitian operators
Does every Hermitian operator correspond with an observable?
Yes
What makes an operator Hermitian?
It is equal to its Hermitian conjugate
Can operators between a bra and a ket act on the bra before it, the ket after it or either?
Either
Is the position operator Hermitian?
Yes
What are eigenstates (or eigenvectors) of an operator?
A set of non-zero vectors associated with each operator and when the operator acts on these vectors, it equals the same vector multiplied by a scalar
What is the basic eigenstate equation with an operator?
The operator acting on an eigenstate is equal to the corresponding eigenvalue multiplied by the same eigenstate
How do you find the eigenvalues?
In matrix form, solve the quadratic where the determinant of the operator minus the eigenvalue multiplied by the identity matrix is all equal to 0
If a vector is an eigenstate of an operator, will its eigenvalue be the same or different from the same vector multiplied by a complex number?
The same
Since any complex number multiplied by an eigenstate corresponds to the same eigenvalue, what will we assume about the norm of the eigenstates to make it easier?
They have unit norm
What is true for all eigenvalues of Hermitian operators?
They are real
Eigenstates of a Hermitian operator corresponding to different eigenvalues are what to each other?
Orthogonal
What type of operator can we find a set of eigenstates which form an orthonormal basis for all of Hilbert space and what is this set of states called?
Hermitian and an eigenbasis
What is an eigendecomposition of a Hermitian operator?
The sum over the eigenbasis of the operator of the eigenvalue multiplied by the corresponding eigenstate in the basis multiplied by its corresponding dual vector
A Hermitian operator is what in the basis of its eigenstates? (easiest to think in matrix form for this question)
Diagonal
What is an eigenvalue said to be if it has several different orthonormal eigenstates corresponding to it?
Degenerate
What are the operators called that group together eigenvectors in an eigenbasis with the same eigenvalues?
Projection operators or projectors onto the eigenspace with with the particular eigenvalue
What type of operators are projection operators and what condition do they satisfy?
They are Hermitian and satisfy the condition of them being equal to the square of itself
What is the spectral decomposition of any Hermitian operator?
It is the sum of the eigenvalues multiplied by their corresponding projection operator onto the eigenspace with that eigenvalue
What is the spectrum of the operator?
The collection of all the different eigenvalues of the operator
For any function of a Hermitian operator, what are the eigenstates and eigenvalues?
The eigenstates are the same as the eigenstates of the operator and the eigenvalues are given by the function acting on the eigenvalues of the operator
If an operator in an infinite dimensional Hilbert space have a countable basis of eigenvectors and a corresponding countable set of eigenvalues, what can we call the spectrum and do the results so far also apply here?
Discrete and yes
Is the position state in the Hilbert space and why?
No because they have infinite norm
Do operators in an infinite dimensional Hilbert space without a discrete spectrum technically have any eigenstates?
No
If we treat the position state as if it were an eigenstate of the position operator, what would its eigenvalue be?
x
How can we write any Hermitian operator with a continuous spectrum on the space of wavefunctions?
The integral between minus and positive infinity of the function of ‘a’ multiplied by the vectors of ‘a’ and the dual vectors of ‘a’ d’a’
When an observable is measured, what are the possible outcomes?
The eigenvalues of the corresponding Hermitian operator
What is the probability of obtaining a certain eigenvalue when the operator is measured on the state (not the calculation part)?
The bra of the state multiplied by the projector onto the eigenspace of that eigenvalue multiplied by the ket of the state
What is the probability of a non-degenerate eigenvalue?
The absolute value of the inner product between the corresponding eigenvector of the eigenvalue and the state all squared
What is the probability of a degenerate eigenvalue?
The sum over all states in the eigenbasis with that particular eigenvalue of the absolute value of the inner product between the corresponding eigenvectors of the eigenvalue and the state all squared
What is the sum of all the projectors given a complete eigenbasis?
The identity operator
What is the sum of the probabilities of all eigenvalues of an operator?
1
When is the only time when a measurement will give you a definite outcome?
The state that is made by orthonormal basis of the Hilbert space is an eigenstate of the operator and the outcome of the corresponding eigenvalue will be obtained with certainty (rare)
What is the expectation value of an observable?
The average value which is obtained in the measurement and it is the sum over all eigenvalues of the eigenvalues multiplied by their probabilities
What is another method to calculate the expectation value?
The bra of the state multiplied by the operator multiplied by the ket of the state
How do we ensure that if we immediately repeated exactly the same measurement, we would get the same result?
After a measurement, the states collapses to an eigenstate of the observable with the obtained eigenvalue
If the observed eigenvalue is non-degenerate, what will the state collapse into?
The corresponding eigenstate
If the observed eigenvalue is degenerate, what will the state collapse into?
The projector corresponding to that eigenvalue multiplied by the ket of the state divided by the norm of the same thing (the square root of the probability of that eigenvalue)
How do we know that globular phase factors are physically irrelevant?
They give the same outcome probabilities for all measurements and hence are physically identical
What is the norm of the projector corresponding to an eigenvalue multiplied by the ket of the state equal to?
The square root of the probability of that eigenvalue
What is the expectation value for an operator with a continuous spectrum?
The same as the finite case, with the bra of the state multiplied by the operator multiplied by the ket of the state
What do we have to change for the probability of obtaining a particular outcome of an operator with a continuous spectrum and why?
The probability for the outcome to be in a finite range because the probability of obtaining any particular outcome is zero
What is the projector onto a specified range of positions for an operator with a continuous spectrum?
The integral between the range of positions dx multiplied by the eigenstate and its corresponding dual vector
How do we calculate the probability over a specified range for the position operator (continuous spectrum)?
The bra of the state multiplied by the projector onto the specified range multiplied by the ket of the state, which is equal to the integral between the range of dx multiplied by the normal of the wavefunction (inner product between x and the wavefunction state) squared
What does the square of the norm of the wavefunction represent for position measurements on the state?
The probability density
What is the state after a measurement of position (continuous spectrum)?
A very narrow wavefunction centred on the observed value of position
What Hermitian operator governs the time evolution of a system?
The Hamiltonian
For Hamiltonian evolution, does the inner product between two vectors change with time?
No, it is invariant in time
The norm of a vector is independent is independent of time, what does this mean for normalised states in time?
Normalised states remain normalised as they evolve
What does a time evolution operator do and what letter usually represents it?
It transforms any initial state at time 0 into its corresponding final state at time t and is usually written with a U
Is time evolution in quantum theory linear and why?
Yes because it follows from the linearity of the Schrodinger equation
Is the time evolution operator Hermitian?
No, in general
What does it mean that the time evolution operator in unitary? (This means time evolution in unitary in quantum theory)
The operator multiplied by its adjoint (or vice versa) is the identity operator. Also the adjoint is equal to the inverse of it.
What does it mean that part of unitarity means that time evolution is reversible?
Given some evolution, the inverse transformation (its adjoint) is also unitary and can be used to recover the initial state
Even though it can be difficult to do so, any unitary transformation can be generated by choosing what?
An appropriate Hamiltonian
If we use a time-independent Hamiltonian that is a fixed operator, what physical case does this match with?
A closed system, not subjected to external forces
What can we find if we use a time-independent Hamiltonian?
We can solve Schrodingers equation to find a solution for the time evolution operator, which is e to the -i over h bar multiplied by the Hamiltonian multiplied by time
How do we find the time-evolved state if we have the time evolution operator for a fixed Hamiltonian operator?
Use the time evolution operator on the state at time 0, put the operator in the exponential form with time-independent Hamiltonian (H). Then expand H into its eigenbasis. If we know the decomposition of the initial state in the energy eigenbasis then sub that in.
What happens in the special case where the system starts off in an eigenstate of the Hamiltonian?
The state at later times is the initial state with a phase factor, so it is physically indistinguishable from it and this type of state is called a stationary state
Since energy eigenstates of a time-independent Hamiltonian are all stationary states, if a system starts out in an energy eigenstate, what can we say about it in the future?
It will stay in that state forever with certainty
What observable does the Hamiltonian operator correspond to?
Energy
What is the spin of a particle?
Its intrinsic angular momentum
What did the apparatus made me Stern and Gerlach do to a beam of incoming atoms or molecules?
Deflect them according to their magnetic moment
Why would the particles in the Stern and Gerlach experiment be deflected based off their spin?
Because the spin creates a magnetic moment, which the apparatus deflects
In the Stern and Gerlach experiment, how many deflected beams would they have expected to have seen and how many did they see?
They would expect to see a continuum of outcomes but there was only 2 outgoing beams
What type of operator are the spin components?
Hermitian
What are the eigenvalues for each of the x, y and z spin operators?
+ plus or minus h bar over 2
What are the eigenvectors of each of the x, y and z spin operators?
The up and down vectors in that particular direction
How many dimensions is the Hilbert space for spin-half particles?
2
Can spin be changed the same way normal orbital angular momentum?
No
How many dimensions is the Hilbert space for a particle of spin s and how many equally spaced outgoing beams will there be?
(2s + 1) for both
The eigenstates of each of the spin-half Hermitain operators form what?
An orthonormal basis for Hilbert space
How can we use the Stern-Gerlach apparatus to prepare a beam of atoms that are all in a given state?
Block one of the beams that you don’t want, then after the measurement with the other beam, the state of the spin must be in an eigenstate of spin corresponding to the eigenvalue just measured
What is the spin up state in x direction written in terms of spin up and down in z direction?
square root one over two multiplied by the sum of the spin up and spin down vectors in the z direction
What is the spin down state in x direction written in terms of spin up and down in z direction?
square root one over root two multiplied by the difference between the spin up and spin down vectors in the direction
What is the spin up state in y direction written in terms of spin up and down in z direction?
square root one over two multiplied by the sum of the spin up vector and i times the spin down vector in the z direction
The probability calculations only work for a normalised initial state, if it has a squared norm N, what should we do?
Normalise the state by multiplying it with 1 over root N then calulcate the probabilities
How are spin operators written in matrix form?
It is h bar over 2 multiplied by one of the Pauli matrices
If we square any of the individual spin operators, what is the result?
h bar squared over 4 multiplied by the identity operator
What is the total spin operator and what is it equal to?
The sum of all three spin operators individually squared and it is equal to three quarters times h bar squared times the identity operator
What is the eigenvalues and eigenstates of the total spin operator?
Every spin state is an eigenstate with the same eigenvalue of three quarters multiplied by h bar squared
What is it called that the spin operators of different types follow the uncertainty principle?
They are incompatible observables
When two operators are compatible, what do they share?
A common eigenbasis
What does it mean for measurements if two operators are compatible?
Measuring one operator cannot disturb the results of the other
What is the commutator of A and B?
AB - BA
What can we say about two operators if their commutator is zero?
They commute and they are compatible
What is the commutator of the position and momentum operator and does this make them compatible or incompatible?
i times h bar. Incompatible
For two incompatible operators, A and B, what is the general form of the uncertainty relation?
The standard deviation of A multiplied by the standard deviation of B is greater than or equal to half the norm of the expectation value of the commutator of A and B
How do you calculate the standard deviation of an operator?
The square root of the expectation value of the square of the operator minus the expectation value of the operator squared
What is the uncertainty principle for the position and momentum operators?
The standard deviation of the position operator multiplied by the standard deviation of the momentum is greater than or equal to half times h bar
What is the relation between the commutator of A and B and the commutator of B and A?
They are the negative of each other
How is the commutator linear?
The commutator between A+B and C is equal to the commutator of A and C plus the commutator of B and C
What is the commutator between an operator and a scalar equal to?
0
What is the commutator between an operator and a function of the same operator equal to?
0
What is the shared common eigenbasis of the position operators for X, Y and Z and what does this mean?
Position states r, and it means they are compatible
What is a set of commuting operators that is large enough to define a joint eigenbasis with no joint degeneracies called?
A complete set of commuting observables
What are the central observables R and P in 3D (the ones that are in vector notation but are not vectors in Hilbert space)?
In each there are of 3 components of position or momentum operators for each direction
What are the eigendecompositions in 3D of position (and similarly momentum)?
The integral of d 3 r multiplied by the vector r multiplied by the ket then the bra of r
What is the angular momentum operator in 3D?
The position operator crossed with the momentum operator in 3D (3 components of each)
When rewriting the Hamiltonian for the harmonic oscillator, are the new operators a and a dagger Hermitian?
No
What is the Hamiltonian in terms of a and a dagger for the harmonic oscillator?
h bar multiplied by the angular frequency all multiplied by the sum of a times a dagger and one half
What is the commutator between a and a dagger for the harmonic oscillator?
1
If we have an eigenstate of the hamiltonian, how can we construct a different eigenstate with energy eigenvalue that is h bar times angular frequency higher?
Multiply the a dagger operator with the state and that is the new state
If we have an eigenstate of the hamiltonian, how can we construct a different eigenstate with energy eigenvalue that is h bar times angular frequency lower?
Multiply the a operator with the state and that is the new state
Why must there be a minimum energy eigenstate for the Hamiltonian?
The Hamiltonian is a sum of squared operators therefore it cant have negative energy solutions
What does the a operator multiplied by the minimum energy eigenstate equal?
0
What is the energy of the lowest energy eigenstate of the Hamiltonian?
Half times h bar times by the angular frequency
How are eigenstates of the Hamiltonian labelled?
The number of times an energy of (h bar omega) has been added
When applying the operator a dagger to an eigenstate labelled n, it is equal to a scalar multiplied by the eigenstate labelled n+1, what is the scalar?
root n+1
When applying the operator a to an eigenstate labelled n, it is equal to a scalar multiplied by the eigenstate labelled n-1, what is the scalar?
root n
Why can the operators a dagger and a be considered creation and annihilation operators?
Because they add or subtract packets of energy from energy eigenvalues
When does the harmonic oscillator have the minimal amount of uncertainty allowed by the uncertainty relation between the position and momentum operators?
The ground state (minimum energy eigenstate) labelled with a 0
What is the commutator between the spin operators of different directions?
i times h bar times the spin operator not included in the commutator in the order of x then y then z then back to x etc
What is the commutator between the total spin operator and any of the directional spin operators?
Zero
The commutator relations for spin operators can also be applied for what other type of operators and why?
Angular momentum because spin is a type of angular momentum
What is the difference between the total angular momentum qunatum number (L) and the total spin quantum number (s)?
L can only take integer values, whereas s can be integer or half-integer values
What does it mean that the total spin operator and any spin component operator together form a complete set of commuting observables?
There is no joint degeneracies and can be used to uniquely label a basis in Hilbert space with s and m,|s,m>
What are the quantum number s and m in terms of spin?
Proxy eigenvalues for the total spin operator and the z direction spin operator respectively
What are the eigenvalues of the total spin operator with the eigenstate that is the joint eigenbasis between the total and z spin operators?
h bar squared multiplied by s multipled by (s+1)
What are the eigenvalues of the z direction spin operator with the eigenstate that is the joint eigenbasis between the total and z spin operators?
h bar times m
States with equal values of s (total spin) but different values of m (z direction spin) are connected by what type of operators?
Raising and lowering operators
How do you write the raising spin operator in terms of the x and y spin operators?
x spin plus i times y spin
How do you write the lowering spin operator in terms of the x and y spin operators?
x spin minus i times y spin
How are the spin raising and lowering operators related to each other and what do they both commute with?
They are Hermitian conjugates of each other and both commute with the total spin operator
What is the effect of the raising and lowering operators acting on an eigenstate?
It raises or lowers its quantum number m by one (makes a new eigenstate with an m one more or less than before)
What is the maximum and minimum values of the quantum number m?
s and -s
Why is there a limitation on how many times the raising and lowering operators can be applied to generate new eigenstates?
The possible values for the z-component of spin are limited by the total spin
If you apply the raising or lowering operator on the state with the maximum or minimum values of m respectively, what will it equal?
Zero
Between -s and s, what values does m take?
It takes values from -s to s in integer steps (therefore m can also be integer or half integer depending on what s is)
What is the maximum and minimum value of m for spin-half states?
-1/2 and 1/2
In the individual spin basis for two spins, what are the variables?
s1, m1, s2, m2 or just m1 and m2 if we assume s1 and s2 are fixed
In the combined spin basis for two spins, what are the variables?
s and m
Does the combined total and z direction spin operators commute with the total spin 1 and 2 operators?
Yes
Does the combined total spin operator commute with the z direction 1 and 2 spin operators?
No
What are the 4 operators used for the combined spin basis?
The combined total spin operator, the combined z direction spin operator, the total spin operator for particle 1 and the total 2 spin operator for particle 2
How are the individual basis states related to the combined basis states for two spins?
The sum of the new states multiplied by the relevant Clebsch-Gordan coefficient
Are Clebsch-Gordan coefficients always real and what does this mean?
Yes and it means we can use the same coefficients to transform between the bases in both directions
For two-spin states, why would we want to use the total spin states rather than the individual spin states?
Often the quantum systems are rotationally symmetric
What is a good quantum number?
A quantum number belonging to an observable which commutes with the Hamiltonian
What does the variational principle method do?
It gives an upper bound for the ground state energy
What is the procedure of the variational principle method?
Guess what the normalised ground state could be and then the true ground state energy is always smaller than or equal to the expectation value of the Hamiltonian in the guessed state
What is the big downfall in using the variational principle to approximate the ground state energy and how can we improve the method to reduce this error?
You could guess the ground state badly, so instead you use a family of guesses, related to each other by a variational parameter. This means the upper bound to the ground state energy can move closer to it as you vary the parameter
For a state of two coupled spin half particles, will the x direction spin operator for particle 1 act on just the first part of the state, the second part or both parts?
Just the first part and leave sthe second part unchanged
Which two directions of spin operators will flip the spin direction and which direction will keep it the same? (note: there will also be coefficients that aren’t included here)
x and y will flip and z will keep them the same
What is a variational guess for the ground state of system with two coupled spin half particles? (for the variational principle method)
sin(theta) up-down state plus cos(theta) down-up state (cos and sin can swap)
What do you call three spin states with equal energy (therefore, degenerate) and what is each spin state within it called?
Triplet of spin states and each state within it is called a triplet state
What do you call a non-degenerate spin ground state?
A singlet spin state
What does the perturbation theory do?
It gives an approximation of any energy eigenstate (not just ground state) and eigenvalues of an observable
When is the only case that we can use the perturbation theory?
The observable of interest is close to an observable whose spectrum we already know (we are mostly looking at Hamiltonians here)
What is the main equation for the perturbation theory for Hamiltonians?
The perturbed Hamiltonian (one of interest) is equal to the unperturbed Hamiltonian plus the perturbation (which is lambda= small real number multiplied by V=potential)
In the perturbation theory, what do we assume about the initial eigenvalue we are considering?
It is non-degenerate (the others can be degenerate)
In the perturbation theory, what can we say about the eigenstates and eigenvalues of the perturbed Hamiltonian?
The eigenstates are similar to that of the eigenstates of the unperturbed Hamiltonian and the eigenvalues are close to the unperturbed eigenvalues
In the perturbation theory, how do we define ‘closeness’ in energy eigenvalues between the perturbed and unperturbed Hamiltonians?
The difference in energy is smaller than the energy to move up to another energy in the unperturbed Hamiltonian
In the perturbation theory, the perturbed eigenstate and eigenvalue can be written as what?
A power series expansion in lambda and using the unperturbed eigenstates and eigenvalues
In the perturbation theory, when can we keep only the first few terms of the expansion and if you keep the first 3 terms for example the energy should be correct to what order?
The power series converges so the perturbation is small and it should be correct to order O(lambda cubed)
In the perturbation theory, how do we ensure the perturbed states are normalised?
The scalar product between different eigenstates of the unperturbed Hamiltonian is zero
In the perturbation theory, what is the first order correction to the energy eigenvalue?
The expectation value of the perturbation potential (V) in the unperturbed ground state
In the perturbation theory, how many terms do we have for the perturbed eigenstate and eigenvalue?
2 for the eigenstates and 3 for the eigenvalues
In the perturbation theory, what is the general form of the perturbed eigenstate and eigenvalue?
It is equal to the unperturbed eigenstate or eigenvalue plus lambda multiplied by the first order correction (for the eigenvalue equation there is also lambda squared multiplied by the second order correction)
When do you use the degenerate perturbation theory?
When the specific eigenvalue you are considering is degenerate
In the degenerate perturbation theory, what does the perturbation do to the original degeneracy?
It can be lifted to make a non-degenerate (or less degenerate) spectrum
In the degenerate perturbation theory, what is the operator of V degen?
The perturbation operator projected into the degenerate energy subspace
In the degenerate perturbation theory, the eigenstates of the operator V degen are the same as what?
The eigenstates of the perturbed Hamiltonian
In the degenerate perturbation theory, what is the perturbed eigenstate equal to?
The eigenstate of the operator V degen (correct to zeroth order in lambda)
In the degenerate perturbation theory, what is the perturbed eigenvalue equal to?
The unperturbed energy plus lambda multiplied by the V degen eigenvalue (correct to first order in lambda)
What other transformations of states that are represented by unitary operators obey the same properties as the unitary time evolution operator?
Spatial translation and rotation
What letters typically represent translation and rotation operators?
For translations, S is the operator, with a as the parameter and for rotations, R is the operator with alpha as the operator
What form are the translation and rotation transformation operators in?
The integral of d^3 r (for 3D) then the ket of the changed parameters and the bra of the original. OR it can be represented as the exponent of a generator if continuous
Are translation and rotation transformation continuous or discrete?
Continuous
What is a generator?
It is a Hermitian operator (an observable), and e to the power of the generator is proportional to a continuous transformation operator
What is the generator of spatial translation, rotation and time evolution?
Momentum is the generator of translations, angular momentum is the generator of momentum and the Hamiltonian is the generator of time evolution
What are two types of symmetry in quantum mechanics?
Symmetry of a state and symmetry of an operator
When is a state called symmetric with respect to a transformation?
The transformation acting on the state equals the state, so the state does not change when you apply the transformation
When is a state invariant under a transformation and why is it invariant?
The transformation acting on a state equals a phase factor multiplied the state and it is invariant because a phase factor is physically irrelevant (doesn’t change probabilities)
When is an operator symmetric under a transformation?
The adjoint of the transformation multiplied by the operator multiplied by the transformation is equal to the operator. OR if the operator commutes with the transformation
For symmetric operators, are the expectation values and outcome probabilities changed or unchanged if the transformation is applied to a state before measuring it?
Unchanged
If an operator is symmetric with respect to a continuous transformation (that can be written as the exponent of a generator), what else must be true?
The operator also commutes with the generator
When is a quantum system symmetric under a unitary transformation?
If the Hamiltonian of the system is symmetric under that transformation, so the Hamiltonian commutes with the transformation
For operators, do symmetric and invariant mean the same or different things?
Same thing
When do we call an observable (Hermitian operator) conserved?
If its expectation is time-independent for any initial state
What does an observable commute with if it is conserved?
The time evolution operator and therefore also the time-independent Hamiltonian (since this is the generator of the time evolution operator)
What is Noether’s theorem?
Every symmetry has an associated conservation law, and vice versa, for every conservation law, there’s an associated symmetry
What is Noether’s theorem applied to the time-independent Hamiltonian?
If an operator generate a symmetry of the Hamiltonian then the operator is a conserved quantity and similarly, if the operator is conserved, it generates a symmetry of the Hamiltonian
What commutations imply each other in the phrase ‘time translation symmetry implies the conservation of energy’?
The Hamiltonian (time-independent) commutes with the time evolution operator which implies and is implied by the Hamiltonian commuting with the Hamiltonian
What is Noether’s theorem in terms of unitary transformation operators and their corresponding generators?
If the transformation operator has symmetry, that means the generator is conserved and vice versa
How do you find the time-dependent expectation value in the Schrodinger equation?
The bra of the time evolved state multiplied by the operator multiplied by the ket of the time evolved state
What is the difference between the Schrodinger picture and the Heisenberg picture?
In the Schrodinger picture, the state evolves in time, whilst the observable remains constant. In the Heisenberg picture, the observable evolves in time, whilst the state remians constant
What is the state in the Heisenberg picture at any time and what is it in the Schrodinger frame?
It is the same as the initial state always for Heisenberg and for Schrodinger, it is the initial state multiplied by the time evolution operator
What is the observable in the Heisenberg picture and the Schrodinger picture?
In the Heisenberg picture, it is the adjoint of the time evolution operator multiplied by the initial operator multiplied by the time evolution operator. It is fixed in the Schrodinger picture
How do you find the time dependent expectation value in the Heisenberg picture?
The bra of the (fixed) initial state multiplied by the Heisenberg (time dependent) operator multiplied by the bra of the (fixed) initial state
What is the Schrodinger equation (which governs the time evolution of the state)?
i times h bar multiplied by the differential with respect to time of the time evolved state equals the time dependent Hamiltonian multiplied by the time evolved state
What is the Heisenberg equation where the Hamiltonian is time-independent (which governs the time evolution of the observable)?
the differential with respect to time of the time evolved observable is equal to i over h bar multiplied by the commutator between the time-independent Hamiltonian and the time evolved operator
If the observable commutes with the Hamiltonian then we know it is conserved, what does this mean in the Heisenberg picture?
The operator is time independent so the differential of the observable with respect to time is zero
What is the other name of the interaction picture?
The Dirac picture
What picture do we use for the time-dependent perturbation theory?
The interaction picture
Why is the interaction picture the intermediate one between the Schrodinger and Heisenberg picture?
Both the state and observable are time dependent, but in Schrodinger, just the state is time dependent and in Heisenberg, just the operator is time dependent
How do we split the Hamiltonian in the interaction picture?
Into a well understood time-independent part (H nought) and a more complicated, maybe time-dependent part V
What is the state in the interaction picture?
e to the (i over h bar multiplied by H nought times time) multiplied by the Schrodinger time evolved state
What is the observable in the interaction picture?
e to the (i over h bar multiplied by H nought times time) multiplied by the initial observable multiplied by e to minus the previous one
What is the differential with respect to time of the observable in the interaction picture?
i over h bar multiplied by the commutator of H nought (time-independent) and the interaction picture observable
What is the evolution equation for the state in the interaction picture? (i times h bar times the differential with respect to time of the state in the interaction picture)
The interaction picture of V multiplied by the state in the interaction picture
What is the interaction picture of V?
V sandwiched between the exponential that we keep seeing (first one positive and the second negative in the exponent)
What two conditions are required to use the time-dependent perturbation theory?
V(t) is a weak perturbation to the Hamiltonian and the absolute value of the perturbation multiplied by time divided by h bar is a lot smaller than 1
What is the time evolution operator generated by the interaction picture perturbation?
U subscript V subscript I. The action of this on the initial state (t=0) equals the interaction picture state at time t
What type of series expansion is needed for the time evolution operator generated by the interaction picture V?
Dyson series expansion
What is Fermi’s golden rule?
The transition rate (probability of making a transition from an initial eigenstate to final eigenstate per unit time) is constant in time, given the perturbation is weak
What is the main use of the time dependent perturbation theory?
Calculate transition rates (probability per unit time) between energy eigenstates of H nought due to a weak perturbation
What do you get as a result from the time-dependent perturbation theory?
The time-evolution of a perturbed state
The time evolution operator is equal to a coefficient multiplied by the time evolution operator generated by the interaction picture perturbation, what is the coefficient?
e to the (minus i over h bar times H nought (unperturbed Hamiltonian) times time
