4. The Postulates of Quantum Mechanics

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4. The Postulates of Quantum Mechanics
4A. Revisiting Representations

• Recall our position and momentum operators R and
  P
• They have corresponding eigenstate r and k
• If we measure the position of a particle
• The probability of finding it at r is
  proportional to
• If it is found at r, then afterwards it is in
  state
• If we measure the momentum of a particle
• The probability of momentum ?k is proportional to
• After the measurement, it is in state
• This will generalize to any observable we might
  want to measure
• When we arent doing measurements, we expect
  Schrödingers Equation to work

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4B. The Postulates
Vector Space and Schrödingers Equation
Postulate 1 The state vector of a quantum
mechanical system at time t can be described as a
normalized ket ?(t)? in a complex vector space
with positive definite inner product

• Positive definite just means
• At the moment, we dont know what this space is

Postulate 2 When you do not perform a
measurement, the state vector evolves according
to where H(t) is an observable.

• Recall that observable implies Hermitian
• H(t) is called the Hamiltonian

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The Results of Measurement
Postulate 3 For any quantity that one might
measure, there is a corresponding observable A,
and the results of the measurement can only be
one of the eigenvalues a of A

• All measurements correspond to Hermitian
  operators
• The eigenstates of those operators can be used as
  a basis

Postulate 4 Let a,n? be a complete
orthonormal basis of the observable A, with
Aa,n? aa,n?, and let ?(t)? be the state
vector at time t. Then the probability of
getting the result a at time t will be

• This is like saying the probability it is at r is
  proportional to ?(r)2

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The State Vector After You Measure
Postulate 5 If the results of a measurement of
the observable A at time t yields the result a,
the state vector immediately afterwards will be
given by

• The measurement is assumed to take zero time
• THESE ARE THE FIVE POSTULATES
• We havent specified the Hamiltonian yet
• The goal is not to show how to derive the
  Hamiltonian from classical physics, but to find a
  Hamiltonian that matches our world

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Comments on the Postulates

• I have presented the Schrödinger picture with
  state vector postulates with the Copenhagen
  interpretation
• Other authors might list or number them
  differently
• There are other, equally valid ways of stating
  equivalent postulates
• Heisenberg picture
• Interaction picture
• State operator vs. state vector
• Even so, we almost always agree on how to
  calculate things
• There are also deep philosophical differences in
  some of the postulates
• More on this in chapter 11

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Continuous Eigenvalues

• The postulates as stated assume discrete
  eigenstates
• It is possible for one or more of theselabels to
  be continuous
• If the residual labels are continuous, just
  replace sums by integrals
• If the eigenvalue label is continuous, the
  probability of getting exactly ? will be zero
• We need to give probabilities that ? lies in some
  range, ?1 lt ? lt ?2.
• We can formally ignore this problem
• After all, all actual measurements are to finite
  precision
• As such, actual measurements are effectively
  discrete (binning)

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Modified Postulates for Continuous States
Postulate 4b Let ?,?? be a complete
orthonmormal basis of the observable A, with
A?,?? ??,??, and let ?(t)? be the state
vector at time t. Then the probability of
getting the result ? between ?1 and ?2 at time t
will be
Postulate 5b If the results of a measurement of
the observ-able A at time t yields the result ?
in the range ?1 lt ? lt ?2, the state vector
immediately afterwards will be given by
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4C. Consistency of the Postulates
Consistency of Postulates 1 and 2

• Postulate 1 said the state vector is always
  normalized postulate 2 describes how it changes
• Is the normalization preserved?
• Take Hermitian conjugate
• Mulitply on left and right to make these
• Subtract

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Consistency of Postulates 1 and 5

• Postulate 1 said the state vector is always
  normalized postulate 5 des-cribes how it changes
  when measured
• Is the normalization preserved?

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Probabilities Sum to 1?

• Probabilities must be positive and sum to 1

Postulate 4 Independent of Basis Choice?

• Yes

Postulate 5 Independent of Basis Choice?

• Yes

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Sample Problem
A system is initially in the state When S2 is
measured, (a) What are the possible outcomes and
corresponding probabilities, (b) For each outcome
in part (a), what would be the final state vector?

• We need eigenvalues and eigenvectors

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Sample Problem (2)
(b) For each outcome in part (a), what would be
the final state vector?

• If 0

• If 2?2

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Comments on State Afterwards

• The final state is automatically normalized
• The final state is always in an eigenstate of the
  observable, with the measured eigenvalue
• If you measure it again, you will get the same
  value and the state will not change
• When there is only one eigenstate with a given
  eigenvalue, it must be that eigenvector exactly
• Up to an irrelevant phase factor

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4D.Measurement and Reduction of State Vector
How Measurement Changes Things

• Whenever you are in an eigenstate of A,
  and you measure A, the results are
  certain, and the measurement doesnt change the
  state vector
• Eigenstates with different eigenvalues are
  orthogonal
• The probability of getting result a is then
• The state vector afterwards will be
• Corollary If you measure something twice, you
  get the same result twice, and the state vector
  doesnt change the second time

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Sample Problem

• We need eigenvalues and eigenvectors for both
  operators
• Eigenvalues for both are ? ½?

A single spin ½ particle is described by a
two-dimensional vector space. Define the
operators The system starts in the state If
we successively measure Sz, Sz, Sx, Sz, what are
the possible outcomes and probabilities, and the
final state?

• It starts in an eigenstate of Sz
• So you get ½ ?, and eigenvector doesnt change

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Sample Problem (2)
If we successively measure Sz, Sz, Sx, Sz, what
are the possible outcomes and probabilities, and
the final state?

• When you measure Sx next, we find that the
  probabilities are
• Now when you measure Sz, the probabilities are

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Commuting vs. Non-Commuting Observables

• The first two measurements of Sz changed nothing
• It was still in an eigenstate of Sz
• But when we measured Sx, it changed the state
• Subsequent measurement of Sz then gave a
  different result
• The order in which you perform measurements
  matters
• This happens when operators dont commute, AB ?
  BA
• If AB BA then the order you measure doesnt
  matter
• Order matters when order matters
• Complete Sets of Commuting Observables (CSCOs)
• You can measure all of them in any order
• The measurements identify ?? uniquely up to an
  irrelevant phase

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4E. Expectation Values and Uncertainties
Expectation values

• If you measure A and get possible eigenvalues
  a, the expectation value is
• There is a simpler formula
• Recall
• The old way of calculating expectation value of
  p
• The new way of calculating expectation value of p

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Uncertainties

• In general, the uncertainty in a measurement is
  the root-mean-squared difference between the
  measured value and the average value
• There is a slightly easier way to calculate this,
  usually

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Generalized Uncertainty Principle

• We previously claimed
  and we will now prove it
• Let A and B be any two observables
• Consider the following mess
• The norm of any vector is positive

• The expectation values are numbers, they commute
  with everything
• Now substitute
• This is two true statements, the stronger one
  says

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Example of Uncertainty Principle

• Lets apply this to a position and momentum
  operator
• This provides no useful information unless i j.

Sample Problem
Get three uncertainty relations involving the
angular momentum operators L
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4F. Evolution of Expectation Values
General Expression

• How does ?A? change with time due to
  Schrödingers Equation?
• Take Hermitian conj. of Schrödinger.
• The expectation value will change

• In particular, if the Hamiltonian doesnt depend
  explicitly on time

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Ehrenfests Theorem Position

• Suppose the Hamiltonian is given by
• How do ?P? and ?R? change with time?
• Lets do X first
• Now generalize

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Ehrenfests Theorem Momentum

• Lets do Px now
• Generalize
• Interpretation
• ?R? evolution v p/m
• ?P? evolution F dp/dt

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Sample Problem
The 1D Harmonic Oscillator has Hamiltonian Calc
ulate ?X? and ?P? as functions of time

• We need 1D versions of these

• Combine these
• Solve it
• A and B determined by initial conditions

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4G. Time Independent Schrödinger Equation
Finding Solutions

• Before we found solution when H was independent
  of time
• We did it by guessing solutions with separation
  of variables
• Left side is proportional to ??, right side is
  independent of time
• Both sides must be a constant times ??
• Time Equation is not hard to solve
• It remains only to solve the time-independent
  Schrödinger Equation

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Solving Schrödinger in General

• Given ?(0)?, solve Schrödingers equation to get
  ?(t)?
• Find a complete set of orthonormal eigenstates of
  H
• Easier said than done
• This will be much of our work this year
• These states are orthonormal
• Most general solution to time-independent
  Schrödinger equation is
• The coefficients cn can then be found using
  orthonormality

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Sample Problem
An infinite 1D square well with allowed region 0
lt x lt a has initial wave function in the
allowed region. What is ?(x,t)?

• First find eigenstates and eigenvalues
• Next, find the overlap constants cn

• sin(½n?) 1, 0, -1, 0, 1, 0, -1, 0,

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Sample Problem (2)
An infinite 1D square well with allowed region 0
lt x lt a has initial wave function in the
allowed region. What is ?(x,t)?

• Now, put it all together

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Irrelevance of Absolute Energy

• In 1D, adding a constant to the energy makes no
  difference
• Are these two Hamiltonians equivalent in quantum
  mechanics?
• These two Hamiltonians have the same eigenstates
• The solutions of Schrödingers equation are
  closely related
• The solutions are identical except for an
  irrelevant phase