Physics
Fall Modern 2008, Dr. Starovoitova Lecture 11
Review Lecture 10
Matter waves, De Broglie relations Electron diffraction Waves: , , T, and k Fourier series and Fourier integral
11.1. Uncertainty principle
Electron in space: position, momentum, energy, time Heisenberg showed that no matter how accurate the instruments used, quantum mechanics limits the precision when two properties are measured at the same time.
1 kx 2 2 2 p = = k= h/ p h h px 2
11.1. Uncertainty principle
1 t 2
Many excited states are not stable and decay in about t=10-8 s. Thus E=3x10-8 eV
E = hf = h = h 2
h tE 2
11.2. Heisenbergs microscope
Heisenberg pictured a microscope that obtains very high resolution by using high-energy gamma rays for illumination. The microscope can resolve objects to a size of x, which is related to the wavelength of the gamma ray, by the expression: x = / (2sinA) Momentum before: p=h/ 1)Momentum after: p'x + (h sinA ) / ' 2)Momentum after: p''x - (h sinA ) / '' p'x + (h sinA ) / ' = p''x - (h sinA ) / '' p''x - p'x = px ~ 2h sinA / Since x ~ /(2sinA) and px ~ 2h sinA / , x px ~ h
11.3. Implications
I believe that the existence of the classical "path" can be
pregnantly formulated as follows: The "path" comes into existence only when we observe it. In the sharp formulation of the law of causality-- "if we know the present exactly, we can calculate the future"-it is not the conclusion that is wrong but the premise. Heisenberg, in uncertainty principle paper, 1927
11.4. Quantum wave function
A wave function is a mathematical tool used in quantum mechanics to describe any physical system. It is a function a from space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics describe how the wave function evolves over time. Quantum coral shows electron wave functions (sort of)
11.4. Quantum wave function
F(t)
m
Classical Physics: Newtons 2nd Law
r r a = F /m x(t )
Quantum Mechanics: Schrodinger Equation
( x, t )
11.4. Quantum wave function
Statistical interpretation:
( x, t ) dx
2
probability of finding the particle between x and x + dx
= *
2
= 0.2i 0.5 * = 0.2i 0.5
2
Always real and positive!
= (0.2i 0.5)(0.2i 0.5) = 0.04i 2 + 0.1i 0.1i + 0.025 = 0.065
11.5. Indeterminacy
Suppose you measure the position of the particle and find it at point C. Question: Where was it right before the measurement?
C
1) The realist position (at C) - Einstein 2) The orthodox position (wasnt really anywhere) - Bohr 3) The agnostic position (refuse to answer) - Pauli
11.5. Indeterminacy
The position of the particle was never undeterminate, but wa...