8 Responses to “Lecture 2 | Modern Physics: Quantum Mechanics (Stanford)”

1. admin Says:
   January 19th, 2009 at 4:52 pm

   Overview of complex numbers: don’t confuse x and y plane of complex numbers with vector space. Complex numbers do represent a simple vector space, but a 1 dimensional one.

   Therefore z = x +iy, z can be represented as |z> ket vector.
   The complex conjugate z* = x -iy can be represented as

   For multiplying two complex numbers, z₁ z₂, the conjugate (z₁ z₂)* = z₁*z₂*
   Also (z₁*z₂)* = z₂*z₁
   Interchanging z₂ and z₁, i.e. z₂*z₁ ⇒ z₁*z₂

   From this we can see that since *

   If we think of complex numbers as vectors in vector space and multiply them, when you interchange bra and kets its equiv to complex conjugation

   = *

   Complex function: ψ(χ) where x is real but ψ is complex:

   ψ(χ) = ψreal(χ) = iψimag(χ)

   may not have been used to thinking of functions as vectors, but they are vectors in this mathematical sense.

   column vectors are like functions also but with a discrete index (the numbers in the column) rather than a continuous variable, x.

   29:00 Dimension of a vector space: min number of vectors needed so that you can write any vector as a sum with coefficients of that minimum number.

2. admin Says:
   January 19th, 2009 at 5:06 pm

   viewed up to 32:00

3. admin Says:
   January 21st, 2009 at 4:06 pm

   In column vectors number of items gives the dimension.

   a₁
   a₂ ⇒ |A>
   a₃
   a₄
   .
   .
   .

   You need an infinite number of functions to write all functions, therefore functions are an infinite dimensional vector space.

   We can think of functions as vectors in a complex vector space, i.e :

   ψ(x) ⇒ |ψ>

   (⇒ symbol means ‘think of them as’)

   36:00
   The Dual Vector Space. The dual vector space is basically just the complex conjugate vector space (certainly for the purpose of physics). There is a one to one correspondence between the two each vector space comes into being with another vector space that is in a certain sense identical to it - the complex conjugate.

   If A is a vector in the ket space: |A>, then

   There is a mapping between them, for every ket there is a bra:

   |A> ⇔

   (⇔symbol means 'for every')

   these can be complex conjugates rather than bra/kets i.e.:

   ψ*(x) ⇔ ψ(x)

   For notational purposes when we deal with column (ket) vectors, the conjugate (bra) is represented by a row. i.e.:

   a₁
   a₂ ⇔ (a₁* a₂* a₃* a₄*)
   a₃
   a₄

   note you need to conjugate the complex coefficient in α|A>, (you need to conjugate both parts of a complex product) i.e.:

   α|A> ⇔

   but otherwise, for addition:

   |A> + |B> ⇔

   44:00 Thats the end of the basic idea of linear vector spaces and their duals (or conjugates).

   Next idea is the inner product of two vector spaces. This is the analog of ordinary vector dot products.(the dot product is the projection of one vector onto the others axis and then the two multiplied. i.e. ab cos angle between them:

   ab cosθ or in terms of components:

   a . b = asubxbsubx + asubybsuby i.e is the sum of the products of the components, axis by axis.

   48:00

4. admin Says:
   January 24th, 2009 at 6:12 am

   48:00 - 59:00 Inner products

   The inner product of a bra and ket vector:

   has some abstract properties:

   1. ) = β (i.e if you double the length of |B> and not

   2. the inner product of + |C> ] = +

   3. is related to but not identical to *
   note that this is the complex conjugate.

   From this = *
   the inner product of a vector with itself is a number not a vector and is the complex conjugate of itself.
   Numbers which are the complex conjugates of themselves are real numbers (they are their own reflections in the x axis of the complex plane, therefore they lie on the x axis with no complex component).

   Also * = z*z = (x +iy)(x -iy) = x^2 + y^2 is always positive.

   *****

   Some rules:

   the inner product of functions:

   < φ|ψ> = ∫dx φ*(x)ψ(x)

   for col vectors:

   (b*, b*, b*) a
   a
   a
   inner product, is just the sum of the product of the components b*a + b*a + b*a

5. admin Says:
   January 24th, 2009 at 12:05 pm

   1:05 basis vectors

   unit vectors orthogonal to each other to represent other vectors in their terms. the square of their vector length (dot product) is 1. A vector of unit length is called normalized or normal.

   |bsubi> where i is the dimension.

   = 0 (orthogonal)
   = 1 (i.e. if i =j)

   if both normalized and orthogonal, then called ortho normal.

   = δsubisubj (chronica [sic?] delta is shorthand for 1 when i = j or 0 not = j).

   IT shoudl be possible to take any vector V and express it as the sum over the basis vectors i of coefficients (Vsubi) times the i basis vector

   i.e. |V> = ∑ Vsubi |bsubi> for all i

   skipping a couple of steps:
   = Vsubi

   in general

   |V> = ∑ |subi>

   The above crops up a lot.

   *******

   1:15 In classical mechanics the collection of states in phase space form a set and you can do the mathematical operations on the set e.g. intersection/AND, union/OR

   In QM the concept of AND and OR are quite different (will find out more later). The difference derives from the fact that the states are not points in a set but vectors in a vector space. The algebra of vectors is very different from the boolean algebra of set theory

   What is the connection between states in QM and vectors?
   Classical set of coins for example is set of Head H and Tails T
   H and T form a phase space.

   1:17

6. admin Says:
   April 12th, 2009 at 11:37 pm

   What is the connection between states in QM and vector spaces?

   Take a classical phase space of 2 states, say heads and tails. You could consider each to be described by an orthogonal vector, |H> |T>, but it wouldn’t be useful, classical phase spaces are based upon sets and boolean algebra, the notion of things like adding two points together (heads and tails) as vectors to get a third point ( a headytail) wouldn’t mean anything. In quantum theory, a combination (superimposition) of states is meaningful, so they can be combined as vectors.

   Postulate 1: In QM, if you can differentiate between two superimposed states, say αsubh|H> + αsubt|T> with a single observation, then the vector components are orthogonal i.e. <H|T> = 0.

   (If we make |H> and |T> unit length, i.e. <H|H> = <T|T> = 0 we can form a basis vector).

   The mixed state gives the probability of finding one or the other state in observation, given by the coefficients (called the probability amplitude), αsubh or αsubt. Since the probability is a positive real number whereas the probability amplitudes can be positive, negative or complex, the Probability of H, Psubh = αsubh* αsubh
   Because certainty = 1 in probability, αsubh* αsubh + αsubt* αsubt = 1 (which is the same as the probability of the inner product of the headytail vector with itself ).
   Postulate 2: Any properly defined state of a system should have unit length, i.e. be normalized since probabilities should add up to 1.

   ***********************

   Operators
   Operations that you do on vectors. Specifically linear operators, like reflect in the x axis, scale by a number etc. not non-linear operators like square the length etc. (Notation: Bold number or number with a caret on top.)

   Properties:
   1. A linear operator that acts on a vector gives another vector. i.e. L |A> -> |B>
   and L α|A> = α L |B>
   2. L { |A> |B> } = L|A> + L|B>

   *********************

7. admin Says:
   April 12th, 2009 at 11:41 pm

   Linear operators correspond to quantities that can be measured, ‘observables’ rather than states. You can measure the headsness or tailsness of a coin, so the headsness or tailsness is represented by a linear operator.

8. darkwater Says:
   July 25th, 2009 at 4:42 am

   Two possible interpretations of the quantum mechanics .
   The first, the photon able of timetravel.
   Goes somewhere, afterwards goes back in the time-dimension .
   This way the photon can travel on all possible path.

   The second possible interpretation, the vacuum is a grid.
   The grid always filled with inactive photons.
   Doesn’t matter matter , how many photon we see.
   Probability density is real density of particles , always.

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