Ch 3. The Quantum Mechanical Postulates MS310 Quantum Physical Chemistry - Summarized the rules for...

Ch 3. The Quantum Mechanical Postulates

MS310 Quantum Physical Chemistry

- Summarized the rules for how information is obtained - Summarized the rules for how information is obtained from wave functions in a few postulatesfrom wave functions in a few postulates

- Comparison of postulated results from quantum - Comparison of postulated results from quantum mechanics with those obtained from classical mechanicsmechanics with those obtained from classical mechanics


1) What is postulates? framework of Q.M

2) How many postulates in Q.M? 6 postulates

No exception of these postulates until now.

In this chapter, we explain the 5 postulates. In chapter 10, 6th postulate is introduced.

Basic concepts of Q.M : Postulates


Postulate 1 : The state of a quantum mechanical system is completely specified by a wave function Ψ(x,t). The probability that a particle will be found at time t0 in a spatial interval of width dx centered at x0 is given by Ψ*(x0,t0)Ψ(x0,t0)dx.

3.1 Physical meaning of wave function

Meaning of wave function Ψ(x0,t0) in classical waves sound wave : pressure at (x0,t0) water wave : height of water at (x0,t0)

What is the meaning of Ψ(x0,t0) by Schrödinger equation? → probability of finding a particle(also has a wave character) at position x0 , time t0 within an interval dx.

dxtxdxtxtxtxP 2000000

*00 |),(|),(),(),(


Unlike the classical wave, amplitude of Ψ(x0,t0) has no physical meaning in Q.M. Why? → probability P α square of the magnitude of Ψ(x0,t0)

Ψ(x0,t0) : complex function→ can multiply -1 or change the phase by multiplying the complex number eiθ (θ : phase angle)

However, take the square, Ψ*(x0,t0)Ψ(x0,t0), these effects are cancelled.

Therefore, all wave functions with a different phase angle generate the same observable.

1)( ,1)1( 2 ii ee


Ψ*(x0,t0)Ψ(x0,t0) : probability → sum of probability over whole interval must be 1 :

normalization

1),(),(* dxtxtx

Therefore, Ψ*(x0,t0)Ψ(x0,t0) must satisfy the following conditions.

1) single-valued function(only one probability at each point)

2) 1st derivative exists and continuous(2nd derivative exist and well-behaved)

3) no infinite amplitude over a finite interval(wave function must be normalized)


Example of double-valued function and single-valued function

Continuous and discontinuous function

3.2 Every observable has a corresponding operator


Postulate 2 : For every measurable property of the system in C.M such as position, momentum, and energy, there exists a corresponding operator in Q.M. An experiment in the lab to measure a value for such an observable is simulated in the theory by operating on the wave function of the system with the corresponding operator.

All Q.M operator : Hermitian operator(real eigenvalue)

Order of operation is important

xixpx

xxxixxp

xx

x

x

cos)(ˆˆ

)cos(sin)(ˆˆ

sin)( Ex)


3.3 The result of individual measurement

Postulate 3: In any single measurement of the observable that corresponds to the operator Â, the only values that will ever be measured are the eigenvalues of that operator.

Ex) Hydrogen atomMeasured energies in experiment : only eigenvalues of the time-independent Schrödinger equation

This make senses because the energy levels of the hydrogen atom is discrete and only those energies are allowed.

),(),(ˆ txEtxH nnn

3.4 The expectation value


Postulate 4 : If the system is in a state described by the wave function Ψ(x,t), and the value of the observable a is measured once each on many identically prepared systems, the average value(also called expectation value) of all of those measurement is given by

If Ψ(x,t) is normalized, denominator is 1.There are two cases 1) Ψ(x,t) is a normalized eigenfunction of Â 2) Ψ(x,t) is not a normalized eigenfunction of Â

dxtxtx

dxtxAtx

a

),(),(

),(ˆ),(

*

*


1) Ψ(x,t) is a normalized eigenfunction of Â, φj(x,t)

),(),(ˆ

),(),(

txatxA

txtx

jjj

j

jjjjjj adxtxtxadxtxAtxa

),(),(),(ˆ),( **

All measurements will give the same answer, aj

2) Ψ(x,t) is not a eigenfunction of Â

),(),( txbtx nn

Ψ(x,t) is normalized → ∑bm* bm = ∑ | bm |2 = 1

dxtxtxabb

dxtxbatxbdxtxAtxa

nmm n

nnm

nnnn

mmm

),(),(

]),(][),([),(ˆ),(

*

1 1

*

11

***


Use eigenfunctions of Â form an orthonormal set

1

2

1

**

1 1

* ||),(),(m

mmm

mmmnmm n

nnm ababbdxtxtxabba

bm : expansion coefficient of the wave function

<a> : weighted average

| bm |2 : contribution of each eigenfunction to the wave function Ψ(x,t)

No way of knowing the outcome of individual measurement and <a> is only average value.

What happened in this case?

→ superposition state


Superposition state : wave function has characters of ‘more than 2 states’In this case, electron has a 1s, 2s, 2p, 3s character all and probability of each state after measurement is b1

2, b22, b3

2 and b4

2. → probabilistic outcome

Meaning of measurement process : collapse before the measurement : superposition state after the measurement : one state, the measured eigenvalue


φ1(x), φ2(x), φ3(x) : eigenfunctions of Âeach eigenvalue : a1, 4a1, 9a1

)(4

1)(

4

11)(

2

1)(

)(4

11)(

4

1)(

2

1)(

)(2

1)(

4

1)(

4

11)(

3211

3212

3211

xxxx

xxxx

xxxx

Result of individual result : regardless of state(only a1, 4a1, 9a1 in this case)

Probability of each eigenvalue : depends on state(related to square of each coefficient)


measurement process in quantum mechanics:probabilistic ----> deterministic

The act of carrying out a quantum mechanical measurement

appears to convert the wave function of a system to the

eigen-function of the operator corresponding to the

measured quantity!!

3.5 The evolution in time of a Quantum Mechanical system


Postulate 5 : The evolution in time of a quantum mechanical system is governed by the time-dependent Schrödinger equation :

Meaning of this postulate : variation of wave function as time.(can predict the time variation of wave function – state)However, postulate 4 and 5 is not contradictory. Why?

If we measure the system at t0, and no more measurement after t0, wave function follows the postulate 5 at t1> t0. However, if we measure the system at t1 > t0, wave function follows the postulate 4 after t1.

t

txitxH

),(

),(ˆ


If system is time-independent, wave function is given by

tEiextx )/()(),( In this case, we can solve the eigenvalue equation for time-independent operator Â.

),(),()(ˆ

),(),()(ˆ

),(),()(ˆ

)/()/(

txatxxA

etxaetxxA

txatxxA

nnn

tEinn

tEin

nnn


- Quantum mechanics can be formulated in terms of six postulates provided a convenient framework for summarizing the basic concepts of quantum mechanics.

- The state of a quantum mechanical system is completely specified by a wave function Ψ(x,t). The probability that a particle will be found at time t0 in a spatial interval of width dx centered at x0 is given by Ψ*(x0,t0)Ψ(x0,t0)dx.

- For every measurable property of the system in C.M such as position, momentum, and energy, there exists a corresponding operator in Q.M. An experiment in the lab to measure a value for such an observable is simulated in the theory by operating on the wave function of the system with the corresponding operator.

Summary Summary


- In any single measurement of the observable that corresponds to the operator Â, the only values that will ever be measured are the eigenvalues of that operator.

- If the system is in a state described by the wave function Ψ(x,t), and the value of the observable a is measured once each on many identically prepared systems, the average value(also called expectation value) of all of those measurement is given by

- The evolution in time of a quantum mechanical system is governed by the time-dependent Schrödinger equation :

Summary Summary

dxtxtx

dxtxAtx

a

),(),(

),(ˆ),(

*

*

t

txitxH

),(

),(ˆ

Ch 3. The Quantum Mechanical Postulates MS310 Quantum Physical Chemistry - Summarized the rules for...

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