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

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

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

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

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

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)

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

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

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?

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

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

Sample Problem (2)

(b) For each outcome in part (a), what would be

the final state vector?

- If 0

- If 2?2

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

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

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

100

100

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

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

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

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

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

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

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

Ehrenfests Theorem Position

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

Ehrenfests Theorem Momentum

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

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

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

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

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,

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

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