1. Quantum mechanics I Fall 2012
Physics 451
Quantum mechanics I
Fall 2012
Sep 14, 2012
Karine Chesnel

2. Homework Friday Sep 14 by 7pm: HW # 5 pb 2.4, 2.5, 2.7, 2.8
Quantum mechanics
Homework
Friday Sep 14 by 7pm:
HW # 5 pb 2.4, 2.5, 2.7, 2.8
Tuesday Sep 18 by 7pm:
HW # 6 pb 2.10, 2.11, 2.12, 2.13, 2.14
Thursday Sep 20 by 7pm:
HW # 7 pb 2.19, 2.20, 2.21, 2.22

3. No student assigned to the following transmitters
Quantum mechanics
No student assigned to the following transmitters
1E2B2F1A
1E5C6E2C
1E71A9C6
Please register your i-clicker at the class website!

4. Infinite square well Quantum mechanics Ch 2.2
Properties of the wave functions yn:
1.They are alternatively
even and odd
around the center
Excited states
2. Each successive state
has one more node
3. They are orthonormal
Ground state a x
4. Each state evolves in time with the factor

5. Infinite square well Pb 2.4 Pb 2.5 Quantum mechanics Ch 2.2
Particle in one stationary state
Pb 2.5
Particle in a combination of two stationary states
evolution in time?
oscillates in time
expressed in terms of E1 and E2

6. Quiz 7a Could this function be the wave function
Quantum mechanics
Quiz 7a
Could this function be the wave function
of a particle in an infinite square well at a given time? x a
Yes No
Pb. 2.7

7. Decomposition of any wave function
Quantum mechanics Ch 2.2
Infinite square well
Decomposition of any wave function
At time t = 0
Fourier’s series expansion

8. How to find the coefficients cn?
Quantum mechanics Ch 2.2
Infinite square well
How to find the coefficients cn?
Dirichlet’s theorem
Pb. 2.7 & 2.8

9. Expectation value for the energy:
Quantum mechanics Ch 2.2
Infinite square well
Expectation value for the energy:
The probability that a measurement
yields to the value En is
Normalization

10. Quiz 7b Could this function be a solution for the wave function
Quantum mechanics
Quiz 7b
Could this function be a solution for the wave function
of a particle in an infinite square well at a given time? x a
Yes No

11. Quantum mechanics Ch 2.3
Harmonic oscillator x
V(x)

12. Solving the Schrödinger equation:
Quantum mechanics Ch 2.3
Harmonic oscillator
Solving the Schrödinger equation: x
V(x)

13. Harmonic oscillator Quantum mechanics Ch 2.3x
V(x)
Expressing the Hamiltonian
in terms of convenient operators:
Commutator: or

14. Harmonic oscillator Quantum mechanics Ch 2.3 Ladder operators:
If y is a solution the Schrödinger equation for energy E
Then is a solution the Schrödinger equation for energy
Quantization
of energy
And is a solution the Schrödinger equation for energy
We can built all the solutions
just starting from one solution (ground level)

15. Starting from ground state
Quantum mechanics Ch 2.3
Harmonic oscillator
Ladder operators:
Raising operator:
Lowering operator:
Starting from ground state