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PH 401

PH 401. Dr. Cecilia Vogel. Review. Resuscitating Schrödinger's cat Pauli Exclusion Principle EPR Paradox. Sx, Sy, Sz eigenstates spinors, matrix representation states and operators as matrices multiplying them. Outline. Eigenstates of Spin Components.

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PH 401

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  1. PH 401 Dr. Cecilia Vogel

  2. Review • Resuscitating Schrödinger's cat • Pauli Exclusion Principle • EPR Paradox • Sx, Sy, Sz eigenstates • spinors, matrix representation • states and operators as matrices • multiplying them Outline

  3. Eigenstates of Spin Components • Let the eigenstates of Sz be represented by • |+> • and |-> • Sz|+> = /2 |+> • Sz|-> = -/2 |-> • Then the eigenstates of Sx and Sy • are not eigenstates of Sz • but rather linear combinations of |+> and |->

  4. Eigenstates of Sx and Sy • The eigenstates of Sx and Sy • are linear combinations of |+> and |-> • |Sx=+/2> = • |Sx=-/2> = • |Sy=+/2> = • |Sy=-/2> = • Note that the amplitude of |+> and |-> in these equations for the eigenstates, when we take the absolute square, gives us the probability … • that we will find that value of Sz, given that the particle is in that eigenstate of Sx (or Sy)

  5. Matrix notation • Instead of writing all that out, • we can write each as a matrix, listing the amplitudes in the linear combo: • |+> = • |-> = • |Sx=+/2> = • |Sx=-/2> = • |Sy=+/2> = • |Sy=-/2> =

  6. Overlap • The overlap of two states |a> and |b> • is written <b|a> • bra…ket • tells you how much state |a> is like state |b> • can be calculated by multiplying matrices • ket matrix is the column matrix we just saw • bra matrix is the transpose (turn row into column) and complex conjugate • The absolute square of the overlap • is the probability of finding state |b> when observing a particle known to be in state |a>

  7. Overlap • Let’s calculate the overlap of |Sy=+/2> and |-> • < Sy=+/2 |->= • If we take the absolute square of this, we get the probability that a particle known to have Sy=+/2 will be found to have Sz=-/2. That prob=1/2. • Likewise the probability that a particle known to have Sx=+/2 will be found to have Sy=-/2 can be calculated: • < Sy=-/2 | Sx=+/2>= • Prob = • (Note: you’ll get ½ any time the two dir’s are ┴ ) note complex conj

  8. Overlap • Let’s calculate the overlap of |Sy=+/2> and | Sy=+/2> • < Sy=+/2 | Sy=+/2>= • If we take the absolute square of this, we get the probability that a particle known to have Sy=+/2 will be found to have Sy=+/2. • That prob better be 1, or 100% • If you know it has Sy=+/2, then it is in an eigenstate of Sy, and you will find that value 100% of the time • generally: <a|a> = 1 • if state is normalized

  9. Matrix notation • In matrix notation • states are written as column vectors, • operators are written as square matrices. • Operating with an operator on a state • means multiplying a square matrix by a column matrix • … the result is a column matrix, • as it should be: • when you operate on a state, you should get a (probably un-normalized) state

  10. Matrix notation • In matrix notation • the observable operators corresponding to each component of the spin are given by these matrices: • Sx = • Sy = • Sz =

  11. Eigenstates • We can verify that the eigenstates ?I gave earlier are indeed eigenstates with the stated eigenvalue • For example, is |+> and eigenstate of Sz? • Sz |+>= • =/2 times the original state • so it is an eigenstate with eigenvalue /2 • Similarly Sy | Sy=-/2> • = • =- /2 times the original state • so it is an eigenstate with eigenvalue -/2

  12. PAL

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