4 Responses to “Lecture 1 | Modern Physics: Quantum Mechanics (Stanford)”

1. admin Says:
   January 14th, 2009 at 3:33 pm

   12:00 - 13:30 imagine randomness applied to classical mechanics, e.g. a random jolt to moon revolving around earth - this would result in energy not being conserved. QM randomness still allows for conservation of energy.

2. admin Says:
   January 14th, 2009 at 5:26 pm

   14:00
   Three examples of the oddness of QM randomness:

   1. Two slit experiment
   classical view two slits - sum distribution
   actual QM view two slits - interference (even although 1 photon at a time) interference disappears if only one slit open.

   2. Observation
   2 state system: possible evolutions as you observe:
   heads,heads,heads… and tails,tails,tails…
   or
   heads,tails,heads…
   or
   tails,heads,tails…
   3 state system - coin with three sides (head, tails, feet)
   evolution based on some law of physics: cycle of heads,tails,feet - system is deterministic and reversible, i.e. no information lost.
   If you let this system run for a particular period of time, then reverse laws of physics so that it runs in reverse for the same amount of time, then it will return to the same point.
   If you add some classical randomness to this system, then it will not necessarily be reversible to the same point. i.e. information is lost.
   41: In QM systems is reversible as long as it isn’t observed (as in recorded by anything), specifically if it is observed the original randomness, that is in the system, if at all, is compounded.
   This causes QM collapse if which slit is observed.

   47:30 Points out that the overall logic of QM is different.

   52: Heisenberg derived uncertainty mathematically but then showed observation description with the example of bouncing a photon off a particle.

   E = hf = hbar ω, where ω is angular frequency (2πf)

   For non relativistic particles energy = 1/2 pv (p is momentum v is velocity)

   For relativistic particles, energy approaches pv, so energy of photon is pv

   where p = E/c = hf/c
   since velocity c is wavelength (λ) times freq
   p = hf/λ f = h/λ (de broglie equation)
   i.e. smaller wavelength the larger the momentum

   1:02 assume you want a photo of a particle where it is non fuzzy at a scale of Δ x, then wavelength must be less than Δ x
   λ < Δx

   End 1:03

3. admin Says:
   January 15th, 2009 at 4:42 am

   1:06 why is the heisenberg uncertainty diff from classical physics?
   because in classical physics, light does not come in quanta (photons), therefore you can measure with as small an amount as you like.
   huh - even in classical, a small wavelength measurement will surely be inaccurate.

   1:18 classical: collection of states form a set of points in phase space. basic logic of classical physics in phase space is set theory. A point is a state.
   If you have uncertainty about the point in classical physics you don’t have the max knowledge you could.
   In QM states do not form sets, they are not points but rather vectors in a vector space. [why is a vector implied from the probability uncertainty].
   1:25 Specifically a vector space over complex numbers or a Hilbert space.
   (the ordinary use of the term vector as a pointer, an arrow with a length is merely a specific use of the much more general and abstract term vector that we use here, called ket vectors where the equiv of a point a is denoted by |a>
   1:30 Properties of vectors in Hilbert space:
   1. Given a vector you can multiply it by a complex number and you get a new vector: α|a> = |b>
   2. you can add two vectors to get a new vector: |a> + |b> = |c>
   This is very different from classical mechanics where it simply does not make sense to multiply points in a set by a number. e.g. heads and tails - it doesn’t mean anything to multiply heads by three.
   The fact that states are vectors is odd (will be explained later).

   3. 1 and 2 imply that you can add complex number multiplied vectors: α|a> + β|b> = |c’>
   in other words the notion of addition is well defined, just like it is for ordinary vectors over real number space (where you draw the parallelogram for addition).

   2 examples:
   Hilbert originally invented Hilbert space to describe functions:
   For a complex number function of a single variable (i.e. the variable is ordinary number but the function itself can be complex):
   ψ(x) which has a real and complex part i.e. ψ real (x) + iψ imaginary (x) whcih translates to a function where for every point x, there is a complex number ψ of x or equivalently 2 real functions (i.e. the right hand side)

   Assertion 1: The collection of complex functions of x form a vector space over the complex numbers (this would be a particular type of Hilbert space),

   therefore you should be able to multiply each complex function by a complex number and get a new function or add pairs of functions together to get a new one.
   i.e αψ(x) =ψ’(x) and ψ(x) + φ(x) = ψ”(x)
   the above is enough to tell you that complex functions are a vector space over the complex numbers

   (Real functions are a vector space over the real numbers, but you cannot multiply a real function by a complex number and get back a real function (you get back a complex function), so real functions are not a vector space over the complex numbers.

   Assertion2.
   Complex vector space over complex numbers can be n dimensional and represented by column vectors e.g. for 4 dimensional complex vector space:

   asub1
   asub2
   asub3
   asub4

   1:40 - addition of two vectors and multiplication of vector by complex number applies here therefore collection of objects (components of a vector) in a column also forms a vector space of dimension dictated by column length.

   To give a counter example there are no obvious rules for adding column vectors of different dimensions, so they do not form a vector space.

   ****
   Incidentally, complex numbers are themselves a vector space (you can multiply a complex number by a complex number to get new and you can add two together to get new) - they are the one dimensional vector over the complex number space.

   Complex conjugation:
   complex conjugate is a mapping of the number so that it is reflected in the x axis
   i.e. z=x +iy -> x - iy, called z*

   For every vector space there is a notion of a dual vector space.

4. admin Says:
   April 12th, 2009 at 2:39 pm

   Just over a hour into this lecture, Susskind says that Heisenberg uncertainty is different from classical physics because the discrete nature of photons mean that in essence there is a limit to how ‘gently’ you can measure a system.

   But surely, even if a classical measurement can be arbitrarily ‘gentle’ there is still uncertainty between position and momentum, since an extremely low energy, long wavelength measurement would give very inaccurate location when measuring momentum without disturbance, and vice versa?

   I emailed Susskind and got this reply (amazing that he was kind enough to do that):

   “The difference is that in classical physics the intensity (energy) of a light beam of given frequency can be arbitratily weak. But in quantum mechanics is cannot be less than 1 photon.”

   In other words, remember that in the classical view, energy is based on intensity (i.e. amplitude) and is independent of frequency or wavelength.

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