1. Quantum mechanics I Fall 2012
Physics 451
Quantum mechanics I
Fall 2012
Oct 19, 2012
Karine Chesnel

2. Next homework assignments:
Phys 451
Next homework assignments:
HW # 14 due Friday Oct 19 by 7pm
Pb 3.7, 3.9, 3.10, 3.11, A26
HW #15 due Tuesday Oct 23
Pb 3.13, 3.14, 3.17, 3.18, 3.22, 3.23
Practice test 2 M Oct 22
Work with your group!
Test 2 : Tu Oct 23 – Fri Oct 26

3. Generalized statistical interpretation
Phys 451
Generalized statistical interpretation
Particle in a given state
We measure an observable
(Hermitian operator)
Operator’s eigenstates:
eigenvector
eigenvalue
Eigenvectors are complete:
Discrete spectrum
Continuous spectrum

4. Generalized statistical interpretation
Phys 451
Generalized statistical interpretation
Particle in a given state
Fourier’s trick to find Cn
Normalization:
Expectation value

5. If you measure an observable Q on a particle in a certain state ,
Phys 451
Quiz 18
If you measure an observable Q
on a particle in a certain state ,
what result will you get?
the expectation value
one of the eigenvalues of Q
the average of all eigenvalues
A combination of eigenvalues
with their respective probabilities

6. Generalized statistical interpretation
Phys 451
Generalized statistical interpretation
Operator ‘position’:
Probability of finding the particle at x=y:

7. Generalized statistical interpretation
Phys 451
Generalized statistical interpretation
Operator ‘momentum’:
Probability of measuring momentum p:
Pb 3.11: probability of measuring p in a given range

8. Different notations to express the wave function:
Phys 451
The Dirac notation
Different notations to express the wave function:
Projection on the position eigenstates
Projection on the momentum eigenstates
Projection on the energy eigenstates

9. The uncertainty principle
Quantum mechanics
The uncertainty principle
Finding a relationship between standard deviations
for a pair of observables
Uncertainty applies only for incompatible observables

10. The uncertainty principle
Quantum mechanics
The uncertainty principle
Position - momentum

11. The uncertainty principle
Quantum mechanics
The uncertainty principle
Position - Energy
Pb 3.14

12. The uncertainty principle
Quantum mechanics
The uncertainty principle
Energy - time
Special meaning of Dt

13. An excited particle emits a certain radiation of energy E
Quantum mechanics
Quiz 19
An excited particle emits a certain radiation of energy E
with a band width DE. What can we say
about the characteristic lifetime of excited state?
Lifetime is a least
Lifetime is a most

14. Heisenberg equation of motion
Quantum mechanics
Heisenberg equation of motion
Pb 3.17
constant
Ehrenfest’s theorem

15. Heisenberg equation of motion
Quantum mechanics
Heisenberg equation of motion
Definition for Dt:
when
To evaluate Dt:
choose an appropriate operator
calculate and
Pb 3.18
Application: use Q = x, in the case of the infinite square well

16. The Dirac notation Inner product: “bra” “ket” Operator: Pb 3.22
Quantum mechanics
The Dirac notation
Inner product:
“bra”
“ket”
Operator:
Pb 3.22
Projection operator:
for orthonormal basis
Pb 3.23 or