foundations of quantum mechanicsd40t5n/2017_l1.pdf · quantum states a quantum ‘system’ is also...
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Foundations of Quantum Mechanics
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AdminJames Currie
Talks every Thursday 13:00 CG91 (chemistry)
Slides available after the lecture at my profile page:
www.ippp.dur.ac.uk -> The Institute -> Research Staff
Obviously no homework or workshops or exam (yay!)
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Why more talks on QM?• QM is universal: 117 years old, no deviations found
• QM is (too) useful: applications studied a lot at university, not much time for discussing its meaning (or if it has one)
• QM is radical:
“Anyone who is not shocked by quantum theory has not understood it.” [Bohr]
• QM may be incomplete: many outstanding problems that go beyond calculating things… need a better understanding
• QM is interesting (hopefully to be demonstrated by Easter)
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Series Outline1. How to be a Quantum Mechanic
2. Entanglement and decoherence
3. A Gordian knot and Heisenberg’s cut
4. Local hidden realism: Einstein’s “reasonable” solution
5. QM’s classical inheritance
6. Bohmian realism: non-local hidden variables and holism
7. How many cats does it take to solve a paradox?
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What I will not be talking about
Foundations of QM is a huge field; many things will not be covered:
• Nelson’s stochastic QM
• Objective reduction models (GRW, Schrödinger-Newton states)
• Consistent histories “Copenhagen done right” (supposedly)
• Epistemic interpretations (QBism) for a good introduction see [arXiv:1311.5253]
This reflects my personal bias and should not stop you looking into these if you’re interested!
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Some Excellent BooksThe following are some relevant books I recommend:
• Beyond measure: Jim Baggot. Accessible overview
• The Speakable and Unspeakable in Quantum Mechanics: John Bell. A collection of his papers, all classics
• Foundations and Interpretation of Quantum Mechanics: Gennaro Auletta. Advanced and comprehensive
• Decoherence and the Quantum to Classical Transition: Maximillion Schlosshauer. A modern treatment of QM and decoherence. Also a short introduction article [arXiv:quant-ph/0312059]
• The Quantum Theory of Motion: Peter Holland. The most complete text on de Broglie-Bohm theory
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Lecture 1How to be a Quantum Mechanic
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• For a point particle, ‘state’ at any given time is given by particle’s position and velocity (or momentum)
• can be formulated as a point in a 6 dimensional space (see lecture 5)
• the classical state evolves deterministically in time according to Newton’s laws of motion
• ‘Observables’ (properties) are functions of the state, e.g. potential energy, V(x)
Classical StatesBefore we think about QM, what is a classical ‘state’?
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Quantum StatesA quantum ‘system’ is also specified by a ‘state’:
• system’s degrees of freedom reflected in a ‘Hilbert’ space (a type of vector space). e.g. spin 1/2 system has 2-dimensional Hilbert space
• quantum state is just a point in this space, a vector, denoted by
• quantum states have complex-valued components, e.g.
• ‘inner’ or ‘dot’ product of two quantum states tells us how much they overlap in the Hilbert space, gives a complex number
!
!
Isn’t this all a bit abstract? Yes!
|ψ⟩|φ⟩
⟨φ|ψ⟩
H ⇠ C2
| i✓c1c2
◆with c1, c2 2 C
h�| i ⇠ C
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Superpositions and BasesQuantum states are vectors, so we can add them together
!
e.g 2-d example in component form:
!
!
Each Hilbert space is spanned by a set of basis vectors
We can decompose any state into a superposition of orthonormal basis vectors
!
The expansion coefficients tell us how much of is in the basis state
| i = |�i+ |⌘i
✓ 1
2
◆=
✓�1�2
◆+
✓⌘1⌘2
◆=
✓�1 + ⌘1�2 + ⌘2
◆
|ni
| i =X
n
cn|ni
|nicn | i
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WavefunctionsIf we are interested in a state with continuous degrees of freedom, like position, then we expand the state in the basis of position basis states :
!
The expansion coefficient in this continuous basis is a ‘wavefunction’
In this basis, the dot product between two states is just an integral:
!
Could also expand in a continuous momentum basis,
!
A momentum wavefunction is just the Fourier transform of a position wavefunction
|xi
| i =Z
dx (x) |xi
h�| i =Z
dx �⇤(x) (x)
(x)
| i =Z
dp (p) |pi
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‘Observables’Observables (measurables) in classical mechanics are simple functions of the system’s state, i.e. property values, e.g. spring potential energy
!
Measurables in QM are represented by Hermitian linear operators in the Hilbert space, e.g. for 2-d spin 1/2, spin along z-axis is associated with the operator,
!
Each operator has a special set of values associated with it, called ‘eigen-values’, the closest thing to a classical property value, e.g.
!
When we do a measurement, the only possible outcome is an eigenvalue
V (x) =1
2kx
2
Sz =~2
✓1 00 �1
◆
�± = ±~2
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Eigen-statesEigenvalues are definite property values of the state, like in classical mechanics
These are closely associated with a special set of quantum states called “eigenstates”; each eigenvalue corresponds to an eigenstate
!
The action of an observable’s operator on an eigenstate yields an eigenvalue
For any observable, its eigenstates form a complete basis for the Hilbert space, so can expand any quantum state as a superposition of eigenstates
| i =X
n
cn|ni
O|ni = en|ni
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e-e link• If a quantum system is in an eigenstate of an observable then we
can assign a value to that state for the observable
• If we do an experiment we always measure an eigenvalue of the observable’s operator and claim the system is in an eigenstate
!
!
This is the eigenvalue-eigenstate (e-e) link
• crucial to make the connection between quantum state and “reality”
What if the system is not in an eigenstate?
system in an eigenstate
measure an eigenvalue
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ProbabilityIf the system is not in an eigenstate, then can’t assign a value to it for observables
• must be a superposition of eigenstates
!
• After measurement we claim the system is in an eigenstate
• discontinuous jump (collapse) from a superposition to eigenstate
Can’t say which state it will jump to (indeterminate) but can calculate probability of finding the state in the nth eigenstate upon measurement
!
or for a continuous basis like position:
| i =X
n
cn|ni
P (n) = |cn|2
P (x+ dx) = | (x)|2dx
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Expectation valuesIf quantum states jump indeterminately into eigenstates upon measurement:
• can’t predict an outcome for any given measurement
• but can study averages over many measurements
Average value for an observable given by the expectation value for a particular state
!
or in the position basis,
hOi = h |O| i =Z
dx ⇤(x)O(x) (x)
hOi = h |O| i =X
i,j
c⇤i cjhi|O|ji
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CompatibilityIn CM all property values can be defined simultaneously. In QM only eigenstates have definite values
Eigenstates are inherent (‘eigen') to the observable’s operator:
• if operators have different eigenstates then system can’t be in an eigenstate of both simultaneously
• properties cannot be determined simultaneously; such observables are incompatible
Mathematically, compatible operators commute
!
Incompatible observables result in an uncertainty relation for measurements
[A, B] = AB � BA = 0
�A�B � 1
2|hCi |, if [A, B] = iC
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Time evolutionUntil a measurement, quantum states evolve deterministically
!
If the state is an energy eigenstate,
!
then the time dependence is just a global phase
!
and expectation values are constant in time
H| i = E| i
| (t)i = eiEt/~| (t0)i
h (t)|O| (t)i = h (t0)|eiEt/~Oe�iEt/~| (t0)i = h (t0)|O| (t0)i
i~ @@t
| (t)i = H| (t)i
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Example 1Spin 1/2 system:
• Hilbert space 2-d, can write as a column vector. Consider a state,
!
• observable “spin in z direction” represented by operator
!
• this operator has eigenstates with eigenvalues
!
• Probability the state is found in each eigenstate given by
!
Sz =~2
✓1 00 �1
◆
✓ 1
2
◆=
1p10
✓i3
◆
⇢✓10
◆,+
~2
� ⇢✓01
◆,�~
2
�
P (+~/2) = |c1|2 =1
10, P (�~/2) = |c2|2 =
9
10
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expectation value for spin of this state
!
spin in x or y direction are incompatible observables with spin along z, e.g.
!
with eigenstates and eigenvalues,
!
!
yielding the uncertainty relations,
!
e.g. for our state,
hSzi =
✓1
10� 9
10
◆~2= �2~
5
Sx
=~2
✓0 11 0
◆, S
y
=~2
✓0 �ii 0
◆
[Sx
, Sy
] = i~ Sz
, [Sz
, Sx
] = i~ Sy
, [Sy
, Sz
] = i~ Sx
⇢1p2
✓11
◆,+
~2
�
S+x
⇢1p2
✓1�1
◆,�~
2
�
S�x
,
⇢1p2
✓1i
◆,+
~2
�
S+y
⇢1p2
✓1�i
◆,�~
2
�
S�y
�Sx
�Sy
� ~2|hS
z
i
|
�Sx
�Sy
� ~25
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Example 2Infinite square well
• infinite dimensional Hilbert space
• can expand in continuous position basis… wavefunction
• Energy observable in this basis
!
• energy eigenstates and eigenvalues given by,
!
• can expand any wavefunction in eigenstates
L2[0, L]
(x) = hx| i
⇢ n(x) =
1p2sin
✓n⇡x
L
◆,
n
2~2⇡2
2mL
2
�
H(x) = � ~22m
r2
�(x) =1X
n=1
cn n(x)
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consider the superposition at an instant in time
!
probability to be found in nth state,
!
energy expectation value for this state,
!
!
!
!
!
�(x) =1p3 1(x) +
r2
3 3(x)
hEi� = h�|H|�i
=
Zdx �(x)⇤H(x)�(x)
=
Zdx
1
3 1(x)
⇤H(x) 1(x) +
p2
3 1(x)
⇤H(x) 3(x) +
p2
3 3(x)
⇤H(x) 1(x) +2
3 3(x)
⇤H(x) 3(x)
�
=
Zdx
1
3E1| 1(x)|2 +
p2
3E3 1(x)
⇤ 3(x) +
p2
3E1 3(x)
⇤ 1(x) +2
3E3| 3(x)|2
�
=1
3E1 +
2
3E3 =
19~2⇡2
6mL2
|h n|�i|2 =
����Z
dx n(x)�(x)
����2
=
����X
i
ci
Zdx n(x) i(x)
����2
= |cn|2
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Time dependence of wavefunction from TDSE
!
solve differential equation with usual methods
!
!
!
the state will evolve in time, e.g.
!
probability it is found in initial state again after a time t,
i~ @
@t (x, t) = H (x, t)
�(x, t) = (x)F (t)
,! (x, t) =1X
n=1
e
�iEnt/~cn n(x)
�(x, t) =1p3e
�iE1t/~ 1(x) +
r2
3e
�iE3t/~ 3(x)
|h�(0)|�(t)i|2 =
����1
3
e�i�Et/~+
2
3
e+i�Et/~����2
=
5
9
+
4
9
cos
✓8~⇡2t
mL2
◆
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Summary• QM systems’ degrees of freedom are encoded in a Hilbert space
• QM states are vectors in this space and add linearly
• observables are linear operators in this space, they do not generally commute
• each operator has a special set of numbers (eigenvalues), values of the observable
• the outcome of a measurement is always an eigenvalue
• eigenvalues are associated with eigenstates via the e-e- link
• incompatible observables cannot share eigenstates, implies uncertainty relations
• we can always decompose any state into a superposition of eigenstates
• measurements are indeterminate but we can calculate averages and probabilities
• the state evolves according to the linear Schrödinger equation until a measurement
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Series Outline1. How to be a Quantum Mechanic
2. Entanglement and decoherence
3. A Gordian knot and Heisenberg’s cut
4. Local hidden realism: Einstein’s “reasonable” solution
5. QM’s classical inheritance
6. Bohmian realism: non-local hidden variables and holism
7. How many cats does it take to solve a paradox?