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