1. 4. The Postulates of Quantum Mechanics 4A. Revisiting Representations
Recall our position and momentum operators P and R
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 aren’t doing measurements, we expect Schrödinger’s Equation to work

2. Vectors Space and Schrödinger’s Equation
4B. The Postulates
Vectors Space and Schrödinger’s 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 don’t 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

3. 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 orthonmormal basis of the observable A, with A|a,n = a|a,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

4. 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 haven’t 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

5. 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

6. Continuous Eigenvalues
The postulates as stated assume discrete eigenstates
It is possible for one or more of these labels 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 <  < 2.
We can formally ignore this problem
After all, all actual measurements are to finite precision
As such, actual measurements are effectively discrete (binning)

7. 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 <  < 2, the state vector immediately afterwards will be given by

8. 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:

9. 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?

10. Postulate 4 Independent of Basis Choice?
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

11. 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

12. Sample Problem (2)
(b) For each outcome in part (a), what would be the final state vector?
If 0:
If 22:

13. 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

14. 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 doesn’t 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 doesn’t change the second time

15. 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 doesn’t change
100%
100%

16. 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:
50%
50%
50%
50%
50%
50%

17. 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 don’t commute, AB  BA
If AB = BA then the order you measure doesn’t matter
Order matters when order matters
Complete Sets of Commuting Observables (CSCO’s)
You can measure all of them in any order
The measurements identify | uniquely up to an irrelevant phase

18. 4E. Expectation Values and Uncertainties
If you measure A and get possible eigenvalues {a}, the expectation value is
There is a simpler formula:
Recall:
The old way of calculating P:
The new way of calculating P:

19. 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:

20. 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:

21. Example of Uncertainty Principle
Let’s 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

22. 4F. Evolution of Expectation Values
General Expression
How does A change with time due to Schrödinger’s Equation?
Take Hermitian conj. of Schrödinger.
The expectation value will change
In particular, if the Hamiltonian doesn’t depend explicitly on time:

23. Ehrenfest’s Theorem: Position
Suppose the Hamiltonian is given by
How do P and R change with time?
Let’s do X first:
Now generalize:

24. Ehrenfest’s Theorem: Momentum
Let’s do Px now:
Generalize
Interpretation:
R evolution: v = p/m
P evolution: F = dp/dt

25. Sample Problem The 1D Harmonic Oscillator has Hamiltonian
Calculate X and P as functions of time
We need 1D versions of these:
Combine these:
Solve it:
A and B determined by initial conditions:

26. 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

27. Solving Schrödinger in General
Given |(0), solve Schrödinger’s 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:

28. in the allowed region. What is (x,t)?
Sample Problem
An infinite 1D square well with allowed region 0 < x < a has initial wave function
in the allowed region. What is (x,t)?
First find eigenstates and eigenvalues
Next, find the overlap constants cn
> assume(n::integer,a>0);sqrt(6)/a^2* integrate(sin(n*Pi*x/a)*(a-abs(2*x-a)),x=0..a);
sin(½n) = 1, 0, -1, 0, 1, 0, -1, 0, …

29. in the allowed region. What is (x,t)?
Sample Problem (2)
An infinite 1D square well with allowed region 0 < x < a has initial wave function
in the allowed region. What is (x,t)?
Now, put it all together

30. 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 Schrödinger’s equation are closely related
The solutions are identical except for an irrelevant phase