Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Exams Summer 2022 > Mathematical Methods for Physics > Flashcards

Mathematical Methods for Physics Flashcards

1
Q

What is the requirement on an operator for superpositions of solutions to be solutions themselves?

A

The operator is linear and the equation is homogeneous

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

What is the requirement on an operator for an ordinary point?
What about a regular singular point?
What are the consequences?

A

P(x) and Q(x) must be finite -> 2 analytic solutions

If P(x) has a simple pole or/and Q(x) has a simple or second-order pole. -> at least one solution, possibly with a pole

Otherwise - no solutions

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

What does it mean for a function to be analytic?

A

Taylor expandable, and convergent within some radius.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

What is the wronskian? How does it allow us to determine whether functions are linearly independent?

A

The wronskian is the determinant of a matrix formed of the functions and their derivatives.

W != 0 for linearly independent functions

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

How can we derive the equation for finding second solutions?

A

Differentiate the wronskian and substitute in the governing equation.
This results in a simple DE for the wronskian that can be integrated.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What is the form of a Sturm-Liouville operator?

What are the conditions?

A
  • (p y’)’ + qy

p and q are real
p > 0 in the domain
Hermitian boundary conditions

For an S-L eigenequation, the weight function must also be real and > 0 in the domain.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

What does it mean for an operator to be hermitian (self-adjoint)?
What are the boundary conditions required for this to be the case?

A

integral (v* L u) = [ integral (u* L v) ]*

Where v and u are solutions.

[p v’ u - p u’ v] (between a and b) = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

What are some of the key properties of an S-L eigenproblem?

A

Real eigenvalues
Real and orthogonal eigenfunctions
Eigenfunctions form a complete basis
There is a minimum eigenvalue ( >0 )

Eigensolutions can be classified by the number of nodes in the domain. yn has n-1 nodes.
Eigenvalues ordered this way are increasing

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

What is the form of the wronskian for an S-L operator?

A

W = W_o / p

where W_o is constant

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

S-L eigenfunction degeneracy:

  • for separated b.c’s?
  • for periodic b.c’s?
A

No degeneracy for separated b.c’s

Degeneracy 2 is allowed for periodic b.c’s

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

What are the defining properties of an inner product?

A

Linear
Symmetric
Positive definite

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

What functions arise as the solutions to the radial part of wavefunctions in an infinite circular well?

A 
Bessel functions
(we have to introduce an integrating factor)
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

What is an integrating factor?

How can its form be derived?

A

An integrating factor, w, allows us to cast an equation in the form of an S-L equation.

We need w p_1 = (w p_o)’
this allows us to find a DE for w, which can be easily integrated.

We must check that the S-L FORM satisfies the conditions of an S-L equation.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

What functions arise as the eigenstates of a QHO?

A

Hermite polynomials.

making some absurd ansatz

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

What is the physical motivation for the generating function for legendre polynomials?

A

The electrostatic potential for a point charge.

The solutions satisfy laplace’s equation, which has a general solution expressed in terms of legendre polynomials.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

What is the physical motivation for the generating function for bessel functions?

A

The 2D wave equation has general solutions in terms of bessel functions, and this can be equated to the plane wave solution.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

How can we find whether a series solution converges or not?

A

The ratio test. The ratio of a subsequent term over the current one should be < 1 (in the limit as n-> inf) for convergence.

However, if the series terminates at some finite term, this series will be a solution.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

What is frobenius’ method?

What is an example of its use?

A

Series solution in the case of expansion about a singularity.
Just like the usual series ansatz, but premultiply by x^s.

e.g: to find series solutions to integer bessel’s equation

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

What is the form of a fourier transform?

What about the inverse? (maths transform)

A

FT(y) = integral{ e^-ikx * y }

FT^-1(Y) = 1/2pi * integral{ e^ikx * Y }

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

What is the FT of the derivative of y w.r.t. x?

What about the second derivative?

A

ik Y

(Y is the FT of y)

Second derivative:
-k^2 Y

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

What is the residue theorem?

A

For a contour integral (anticlockwise) about some poles.

The integral = 2 pi i * (the sum of the residues at the poles)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

What is the form of a laplace transform?

What is it important to note?

A

LT(y) = integral[0, inf] { e^-st * y }

All information is lost for t<0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

What is the LT of 1?

A

1/s

s>0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

What is the LT of t^n?

How can this be shown?

A

n! / s^(n+1)

Prove via induction, starting from n=0, 1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

What is the LT of e^at ?

A

1 / s-a

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

What is the LT of 1 / sqrt (t)?

How can this be shown?

A

sqrt (pi / s)

Let u = sqrt(t) and do the resulting gaussian integral

27
Q

What are the LT’s of sin(at), cos(at), sinh(at), cosh(at) ?

A 
LT(sin(at)) = a/ (s^2 + a^2)
LT(cos(at) = s/ (s^2 + a^2)
LT(sinh(at)) = a/ (s^2 - a^2
LT(cosh(at)) = s/ (s^2 - a^2)
28
Q

What is the LT of a delta function (t-t_o)?

A

e^- (s * t_o)

29
Q

What are the two shift theorems for finding inverse LT’s?

A

F(s-s_o) –> e^(s_o * t) f(t) THETA(t)

e^(-s * t_o) F(s) —> f(t - t_o) THETA(t - t_o)

30
Q

What can help in finding the inverse LT of a fraction involving multiple denominators?

A

Use partial fractions

or convolution theorem

31
Q

What is the convolution theorem for laplace transforms?

A

The same as for FT’s, but with the functions in the integral being multiplied by the corresponding THETA function.

32
Q

What is the LT of the derivative of f w.r.t. t?

A

sF(s) - f(0)

The LT of the second derivative can be easily found by applying this twice.

33
Q

What is the defining equation for a green’s function for an ODE?

A 
The operator(x) acting on the green's function(x, z)
= delta(x-z)
The green's function also satisfies the boundary conditions.
34
Q

For an ODE, how do we use a greens function to find a solution?

A

The green’s function is specific to the operator and the boundary conditions, and gives the solution given some source f.

y = integral over the domain { G(x, z) f(z) } dz

35
Q

When would we use the continuity method to find a gf?

What is the general idea behind using the continuity method?

A

When we have a S-L operator with separated homogenous boundary conditions.

  • find solutions to the homogenous equation that satisfy one b.c. each.
  • Define the gf as comprised of both of these solutions, switching at z, and some normalisation on each side u(z).
  • The gf must be continuous in order to define the second derivative, allowing the normalisations to be expressed in terms of the solutions(z) and some factor A(z).
  • We can determine A(z) by integrating over the infinitesimal point x=z, where the operator acting on the gf gives 1.
36
Q

When would we use eigenfunction expansion to find a gf?

What is the general idea behind using the eigenfunction expansion?

A

If the eigenfunctions can be easily found, however, often the resulting gf is not in a useful form.

-The gf is the sum (or integral for continuous eigenvalues) over all eigensolutions of:

[eigenfunction(x) x eigenfunction*(z)] / eigenvalue

37
Q

In what cases does the eigenfunction expansion method fail?

A

If we include a homogenous solution (zero mode), the gf will be undefined due to a divide by zero error.
We must neglect the zero mode in the gf, and require that the source does not excite the zero mode.
The zero mode can be later added to the solution in any quantity (unknown parameter).

38
Q

How does the method of finding a gf for initial value boundary conditions differ from the continuity method?

A

Not by much, the only difference really is that one of the sides of the proposed gf will not have any boundary conditions (so use a superposition of the two lin. ind. sol. to the homogenous equation), while the other will have two.

This results in causality: at a given time t_0, only the source at times t < t_0 can affect the solution.

39
Q

How can we impose outgoing wave boundary conditions when finding a gf using the continuity method?

A

Using an infinite domain, we can specify the form of the solution at +-infinity. Use these solutions for the two sides of the gf, with some normalisation A(z).

(-ikx for left-moving, ikx for right-moving)

40
Q

What are the differences in using the continuity method to find a gf for a 1st order equation compared to a 2nd order equation?

A

For 1st order, the gf can be discontinuous as we only require the first derivative to exist.
Also, there is only one lin. ind. sol. to the homogenous equation, so we cannot use separated boundary conditions, only periodic or initial b.c’s.

41
Q

What is the defining equation for the gf of a PDE?

How do we find the solution given some source?

A

The operator(R, tau) acting on G(R, tau) = delta(R) delta (tau). Where R = r - r’ and tau = t - t’, exploiting translational symmetry.

The solution can be found by performing the double integral of the G(R, tau) * f(r’, t’) dt’ dr’^(n).
i.e: basically a convolution of G and f over all the variables.

42
Q

What is the result of taking the FT of the del operator acting on the gf?
What about the laplacian operator?

A

del(G) —-> i K G

del^2(G) ——> - K^2 G

K is generally a vector

43
Q

What is the general method for finding a gf for a PDE?

A

Take the spatial FT, then we will have some ODE in time for the FT gf, we then must take the inverse fourier transform.

In some cases we can take the FT in both space AND time (noting that the sign convention on the exponential is opposite for time FT), and then take the inverse FT twice. {(n+1)D FT}

44
Q

What is special about the method of solution of the diffusion equation PDE?

A

The integrals over the spatial dimensions are seperable.

45
Q

What do we see as the solution for amplitude at some point given an instantaneous signal at the origin, for the Wave Equation in 1, 2, and 3D?

A

In 3D, there is only an instantaneous amplitude, that propagates outward at c.
In 2D, there is an “afterglow” effect, where the amplitude propagates out at c, but decays exponentially at any one point.
In 1D, the signal is permanently altered everywhere, propagating outward at c.

46
Q

How can we use the wave equation gf to find that for the helmholtz equation?

A

Helmholtz equation is just the wave equation assuming time dependence e^-iwt (steady-state solutions).

If we use the wave equation gf, with a source of time dependence e^-iwt’, we can perform the time convolution, which will effectively give the equation relating the steady-state solutions to the steady-state source. The helmholtz equation gf can be inferred from this.

Note that the helmholtz equation itself (and thus the gf) should contain no time dependence.

47
Q

What is the hermiticity condition on a kernel?

A

K*(z,x) = K(x,z)

48
Q

In what cases is it likely we will be able to convert an integral equation into a differential equation?

A

The kernel must be able to be removed via differentiation, i.e:

  • the kernel is a low-order polynomial in x
  • the kernel is unchanged by differentiation to some order (e.g: exponential or trig)
49
Q

What is the result of differentiating w.r.t. x an integral (not in x but with the upper limit dependent on x)?

A

the derivative of the upper limit * the intergrand (at x = the upper limit)
(check notes)

We must be careful to use the chain rule in the case where the intergrand is also dependent on x.

50
Q

What method of solution should we use for an integral equation with a displacement kernel?

A

A displacement kernel is dependent only on the difference of x and z, so lends itself easily to the form of a convolution.

For a fredholm equation : use a FT (must have infinite domain)

For a volterra equation : use a LT
(however, in this case we lose information about the solution before some time)

51
Q

What method of solution should we use for an integral equation with a separable kernel?

A

The integral equation can be converted into an NxN matrix equation, for a kernel with N terms.
-to construct this matrix equation, multiply through by h_i and integrate w.r.t. x, where h_i are the components in (z) of the terms of the separable kernel.

It is important to note that we must be careful in the case of a non-invertible (singular) matrix.

52
Q

What is the inverse of a 2x2 matrix?

A

(1/the determinant) * the matrix with the leading diagonal switched and the other diagonal negative.

53
Q

What is meant by a source resolvable function in hilbert-schmidt theory?

A

The function that can be expressed as the kernel operator acting on some source function.
A source-resolvable function can be expressed in the basis of normalised eigenfunctions of the integral operator.

54
Q

What is the form of the resolvent kernel, how do we use it and when can we use it?

A

For a hermitian, square integrable kernel where we choose the eigenfunctions to be orthonormal. (we must also be able to find all the eigenfunctions and eigenvalues of the kernel)

The resolvent kernel is the sum over eigenfunctions of:

( eigenfunction(x) x eigenfunction*(z))/ eigenvalue - general eigenvalue

The resolvent kernel plays the same role as a green’s function, we get solution y via the resolvent kernel acting on the source, in the place of the kernel acting on y.

55
Q

What is an equation for the solution of an integral equation, y, using a Neumann series?

A

y = the sum from 0 to infinity of (lambda * the kernel operator)^n * f

56
Q

What is the form of the nth kernel for a fredholm integral equation?
What about for volterra?

A

fredholm:
K_n = integral w.r.t. z’ between a and b { K[x, z’] K_(n-1)[z’, z]}

volterra:
K_n = integral w.r.t. z’ between z and x { K[x, z’] K_(n-1)[z’, z]}

initially K_2 uses K_1 twice in both cases

** in

57
Q

What is the euler equation?

What is the second form and when is this valid?

A

d/dx ( dF/dy’ ) - dF/dy = 0

if the intergrand of the functional, F, has no explicit x dependence (the independent variable), then we may use the second form:

F - y’( dF/dy’ ) = c

These equations are homogeneous!

58
Q

If we intend to extremise some functional I, subject to constraint J - j_0 = 0, what should we do?

A

We extremise I + lambda(J - j_0)
where lambda is an undetermined lagrange multiplier.

i.e: apply the euler equation to F + lambda G, where F and G are the intergrands of functionals I and J respectively.

This results in an extra unknown parameter (so three in total), which can be fixed by requiring the initial constraint to be satisfied.

59
Q

What is the difference in our method of solution of a constrained minimisation problem, if one of the endpoints is not fixed, but subject to some more complex constraint?

A

The euler equations are solved as usual, so the form of the solution will be the same as if the endpoint was fixed.

The difference will come in fixing the constants of integration. There is a more complex equation that must be valid at the unfixed endpoint, given in the formula sheet.

Note that for chains attached to a rod (frictionless), the chain will approach the rod at right angles. (i.e: y’ = negative reciprocal of the rod gradient)

60
Q

What are the special properties of a fredholm integral eigenvalue problem with a finite and hermitian kernel?

A

The eigenvalues are real and the eigenfunctions are orthogonal.

61
Q

What are the shapes of sinh and cosh graphs?

A

Check word document

62
Q

What functional and constraint should we use to give an S-L equation via the rayleigh-ritz variational technique?

A 
F = p y'^2 + q y^2
G = rho y^2 = 1 (ensures finite solution)
63
Q

How can we prove that S-L eigenproblems have a minimum eigenvalue?

A

Use LAMBDA = I / J

where I and J are defined as usual for an S-L eigenproblem.

64
Q

What is the sum of an infinite geometric series a r^n ?

A

a / (1-r)

Exams Summer 2022 (5 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions