1. Ch 3. The Quantum Mechanical Postulates
- Summarized the rules for how information is obtained from wave functions in a few postulates
- Comparison of postulated results from quantum mechanics with those obtained from classical mechanics
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Basic concepts of Q.M : Postulates
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.
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3.1 Physical meaning of wave function
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.
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.
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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.
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Ψ*(x0,t0)Ψ(x0,t0) : probability
→ sum of probability over whole interval must be 1 :
normalization
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)
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Example of double-valued function and single-valued function
Continuous and discontinuous function
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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
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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 atom
Measured 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.
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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
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1) Ψ(x,t) is a normalized eigenfunction of Â, φj(x,t)
All measurements will give the same answer, aj
2) Ψ(x,t) is not a eigenfunction of Â
Ψ(x,t) is normalized → ∑bm* bm = ∑ | bm |2 = 1
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Use eigenfunctions of  form an orthonormal set
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
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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 b12, b22, b32 and b42. → probabilistic outcome
Meaning of measurement process : collapse
before the measurement : superposition state
after the measurement : one state, the measured eigenvalue
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φ1(x), φ2(x), φ3(x) : eigenfunctions of Â
each eigenvalue : a1, 4a1, 9a1
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)
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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!!
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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.
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If system is time-independent, wave function is given by
In this case, we can solve the eigenvalue equation for time-independent operator Â.
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Summary
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.
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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 :
MS310 Quantum Physical Chemistry