Section 3: The Schrodinger Equation & QM Formalism Flashcards
(107 cards)
1
Q
If two wavefunctions Ψ1 and Ψ2 are both solutions of a wave equation which is linear and homogeneous, the superposition principle ensures that
A
c1Ψ1+c2Ψ2 is also a solution of the same wave equation
this can be extended to as many waves as we want
2
Q
a stationary state satisfies the condition
A
|Ψ (x,t)|^2 = |Ψ (x,0)|^2
3
Q
derivation of the 1d time dependent schrodinger eqn
A
- plane wave in form of px-Et / h bar
- consider t=0
- find 1st and 2nd derivs
- compare using energy for free particle
4
Q
since the one-dimensional time-dependent Schrödinger equation for a free particle is linear and homogeneous
A
a linear superposition of plane waves such as wavepakcets will still be a solution
5
Q
We can generalise the Schrödinger equation to the case where a particle is not free, but it is subject to
A
a time-dependent potential U(x)
hence it is acted on by a force F=-dU/dx
6
Q
Whilst we are considering a time-independent potential here, we can have an even more general form of the Schrödinger equation with
A
a time-dependent potential U(x,t)
7
Q
the TDSE contains a first order time derivative which means that f the initial value of the wavefunction is given at t=t0, then
A
the value of the wavefunction at any later time t can be found by solving the TDSE
8
Q
operator
A
mathematical entities that transform one wavefunction into another
9
Q
the tern i h bar d/dt (LHS of TDSE) applied on the wavefunction is
A
the total energy operator
10
Q
the total energy operator
A
a mathematical function, of time t, acting on the wavefunction that returns the total energy and we indicate with the symbol Ê
11
Q
Hamiltonian operator
A
Ĥ yields the same result as the total energy operator but acts on spatial coords instead of time
12
Q
Ĥ is the sum of
A
the potential and kinetic operators
(T hat and U hat)
13
Q
momentum operator in 1D (p hat)
A
RHS of TDSE
-ih bar d/dx
14
Q
Hamiltonian operator =
A
-h bar ^2/2m d2/dx2 + U(x,t)
15
Q
if the potential U(x) is a continuous function of x, then
A
every function in the TDSE must be continuous
16
Q
if U(x) shows any finite discontinuities with x then
A
d2Ψ (x)/dx2 should also have corresponding finite jumps
for this to happen, need dΨ/dx to be continuous with x
17
Q
the continuity of dΨ/dx on the other end also implies that
A
Ψ(x,t) and dΨ/dt have to be continuous
18
Q
if U(x,t) is a continuous function of t, so will
A
Ψ (x,t) and dΨ/dt
19
Q
if U(x,t) has a finite jump with time
A
dΨ/dt will also have a corresponding finite jump, while Ψ (x,t) will still be a continuous function of t
20
Q
how to generalise the 1D TDSE to 3d
A
using r instead of x
and ∇ instead of d/dx2
21
Q
∇^2 =
A
d2/dx2 + d2/dy2 + d2/dz2
(the laplacian operator)
22
Q
gradient of a function f
A
∇f = (df/dx, df/dy, df/dz)
23
Q
how is the probability conserved?
A
consider what happens to probability over time
since no loss of information, expect this to be unchanged
wavefunction can be normalised at all times
24
Q
how to mathematically write down that probability is conserved
A
d/dt integral over v
P(t,r)dr =0
25
how to derive the probability current or probability flux
take LHS of probability conservation
apply TDSE
use definitions of grad and div
introduce vector field
26
divergence theorem (aka Green's theorem)
the integral of the divergence of a vector over a volume V is equal to the surface integral of the component of that vector along the outward normal, taken over a closed surface S
27
Conservation equation for the probability
d/dt P(r,t) = -∇ . J(r,t)
Physically this means that any change in probability density at a point is balanced by a flow of probability current into or out of that (differential) region.
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the term ∇Ψ (r) in J(r) means that
the spatial wavefunctions must be continuous to avoid delta function spikes in flux
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∇.J in the probability conservation means that
the wavefunction must be differentiable everywhere
dΨ/dx must be continuous otherwise there would be an unphysical divergence of dP/dt
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square-integrability coming from the normalisation condition means that
Ψ (x) approaches 0 for |x| approaching infinity
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caveat to the implications for wavefunctions
for infinite potentials and infinitely fast changes, the second rule no longer holds (∇.J in the probability conservation)
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the probability current is identically zero if either
1. the wavefunction Ψ (r,t) is real-valued
2. it has a complex phase which applies uniformly to all space positions, hence unaffected by the space derivatives
33
why is the wavefunction being real-valued trivial
Ψ* = Ψ
(and ∇Ψ * = ∇Ψ )
so the two terms in J cancel
34
complex phase which applies uniformly to all space positions
x, hence unaffected by the space derivative
how is this zero?
if Ψ=e^iΦψ(r)
then Ψ*∇Ψ = e^iΦ e^-iΦψ∇ψ
=ψ∇ψ = Ψ∇Ψ*
and the two terms cancel
35
stationary states, whose time-evolution is a uniform, energy-coupled phase e^iEt/h bar
have no probability flux
they are stationary states precisely because their probability distribution is time-invariant
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TISE derivation
potential time-independent so separation of variables
rewrite the TDSE
divide by Ψ=ψT
constant = E
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TISE rewritten using operators
H ψ(x) = Eψ(x)
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stationary state
state with a defined total energy E and the form
ψ(x,t) = ψ(x)e^-iEt/h bar
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The state (i.e. the wavefunction) itself is evolving with time, but the time evolution is
just a phase
hence the probability density is time dependent
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the hamiltonian operator and the momentum operator and examples of
physical observables
ie physical quantities that we can measure
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In general, observable such as momentum, position, spin, angular momentum, etc, are described by
operators
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Not all operators are
observable
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Observables are represented by
Hermitian operators
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hermitian operators
operators that return real eigenvalues
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the canonical operators in position-space representation are
total energy
position
momentum
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from the canonical operators
all other operators can be constructed
including: kinetic energy, hamiltonian, angular momentum
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consider a matrix A acting on a vector v
if Av=λv
this is an eigenvalue equation
v is an eigenvector of A
λ is the eigenvalue
48
what does the eigenvalue equation mean?
he application of the matrix A on the vector returned the vector
(in the same direction), but rescaled, with a scaling constant λ
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comparing eigenvalue equation with TISE
instead of matrix A and vector v, have operator H and a wavefunction Ψ(x)
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the TISE tells me that if I apply the Hamiltonian operator H on teh wavefunction Ψ(x), i get
the same wavefunction Ψ(x) but rescaled by a real number which is given by the energy E
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in analogy with eigenvalues and eigenvectors, we can say that in the TISE
Ψ(x) is an eigenfunction
E is the eigenvalue
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An eigenfunction Ψ of an operator A is
a wavefunction that respects the eigenvalue equation
AΨ= λΨ
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the action of Ψ does not change the wavefunction itself, it only
rescales it by some scalar value lambda
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normalised eigenfunctions of a Hermitian operator constitute
an orthonormal, complete basis
(so any state could be written using a linear combination of eigenfunctions)
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The eigenfunctions of an operator are linearly independent, meaning that
one eigenfunction can’t be obtained by linear combinations of others. However, they can be used to obtain any other functions,
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The linear independence between two eigenstates Ψi and Ψj means that they are
orthonormal hence their overlap is zero
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the eigenstates Ψ1(x), Ψ2(x),... of hermitian operators are
orthonormal
meaning their projection or overlap of one on the other is zero
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we can write any state as a
weighted sum of eigenfunctions
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Copenaghen interpretation of quantum mechanics
given a state describing a system in a superposition of eigenstates with amplitudes of probability, a measurement of the operator A on the system collapses the wavefunction to one of the eigenstates
returning a quantum number with probability
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copenhagen interpretation - any following measurement on the same system after the collapse to eigenstates will
return eigenvalue ai with 100% probability
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copenhagen - can we recover a superposition state after the measurement
yes - change to another basis of eigenfunctions
(helps to visualise these as vectors)
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a second basis of eigenfunction of another operator can be chosen such that
Ψ=c1'ψ1' +c2'ψ2'
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after the first measurement of Ψ in the ψi basis, the outcome ψi projected onto the new basis ψi' will be
in a superposition ψi=d1ψ1'+d2ψ2'
where the amplitudes di are the projections of the old eigenfunctions ψi on the new basis (see diagram in notes)
64
a sudden change of the potential implies
a sudden change of the hamiltonian
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the expectation value of an operator A calculated on the wavefunction psi is
Ψ*(x) A Ψ(x)
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expectation value - cannot exactly predict outcome deterministically but....
if we run the experiment many times, we can have an average of many measurement, which is what we would expect to obtain in the classical limit
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a way to think about expectation values
make a 'sandwich' of the operator between the wavefunction and its complex conjugate
68
what does it mean to commute
order does not matter
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the expectation value returns the
average of the possible eigenvalues weighted by their probabilities
70
the variance of an operator is defined as
Δ^2A= sigma^2 A = - ^2
useful to demonstrate uncertainty relations
71
variance from TISE
0
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what does it mean that variance is 0?
every time we measure energy on the eigenstate, we find it exactly E
-- stationary states are states with a well-defined total energy
73
stationary states are boring
-well defined energy
-probability not evolving with time
-only have time-evolving global phase
74
stationary states not useless because
the general time dependent solution for any system can be obtained from a weighted sum of stationary states
such a system will have indeterminate energy
75
the TISE is useful because it gives a set of
eigenvalues E1,E2,E3,... and eigenstates ψ1(x), ψ2(x), ψ3(x),... from which we can obtain a total wavefunction for each allowed value of the energy
76
the eigenstates of Hermitian operators are orthogonal, meaning
that the projection or overlap of one on the other is zero
integral is 0 if i does not = j and 1 if i=j
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how can we obtain a general solution to the TDSE
If each of the wavefunctions is a solution of the TDSE, then a linear combination of them is also o solution
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given a set of eigenstates and eigenvalues of the TISE, the general solution of the TDSE is given by
the linear combination
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general solution of TDSE - amplitudes
'weights' for the set of eigenfunctions
80
the general solution of TDSE has indeterminate energy because
the time evolution is not just a global phase now, there is phase-interference arising from the constituent stationary states
81
the order of operators
MATTERS
reflected mathematically in the commutators
82
commutator between two operators is defined as
[A,B] = AB - BA
83
If the commutator between two operators is non-zero, swapping the order of the two operators will result in
different final wavefunctions
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if the commutator is zero,
it does not matter which order we apply A and B
(ie AB=BA)
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properties of commutators
transpose
expansion
jacobi
hermitian conjugate
86
commutator properties - transpose
[A,B] = - [B,A]
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commutators properties - expansion
[AB,C] = A[B,C] + [A,C]B
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commutator properties - jacobi identity
[A,[B,C]] + [B,[C,A]] + [C,[A,B]] =0
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commutator properties - hermitian conjugate
[A,B] † = [B†, A†]
90
an example of non-commutating operators is given by
the conjugate variables x and px
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canonical commutation relation in 1d
[x,px]=i h bar
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canonical commutation relation in 3d
[ri,pi]=i hbar
[ri,pj]=0 for i does not =j
[ri,rj]=0
[pi,pj]=0
93
generalised uncertainty principle for two incompatible operators A and B
ΔAΔb > or = 1/2i |<[A,B]>|
94
This means that the rate of change of an expectation value is proportional to
the commutator of its operator and the hamiltonian operator plus the time-dependence of the measurement operator itself
95
For “normal” time-independet operators, the driver of change in expectation values is whether the
operator A commutes with the hamiltonian
96
if the operator A and the hamiltonian are compatible, then the
stationary states will be eigenstates of A and hence the expectation value will also be stationary
97
the commutation of an operator with the hamiltonian implies
time-invariance
98
Ehrenfest's theorem
the recovery of classical dynamics in the expectation values of quantum mechanics
99
example of Ehrenfest theorem
applying the time evolution fo expectation values to the operators A=x and A=p and using commutation properties we obtain Newton's second law
100
in general, the hamiltonian is the generator of
time-evolution for a quantum system
101
a state of definite energy is an eigenstates of the hamiltonian and its time-evolution is
just a phase, prop to Et/h bar
102
For time-independent potentials, the time-independent Schrödinger equation can be obtained from
the time-dependent one; obtain stationary states by assuming wavefunction separable into
r and t parts.
103
The TISE is an eigenvalue equation, and the corresponding eigenfunctions and eigenvalues are used to obtain
stationary states
104
The general solution for the TDSE can be obtained from a
linear combination of the stationary states with a definite energy.
105
Wavefunctions must be
square-integrable, continuous everywhere, and smooth except at infinite potential steps.
106
Expectation values also time-evolve with the Hamiltonian: static operators which commute with the Hamiltonian have
time-invariant expectation values.
107
