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Lecture I: Dirac Notation • To describe a physical system, QM assigns a complex number (`amplitude’) to each distinct available physical state. – (Or alternately: two real numbers) – What is a `distinct physical state’? • Various common vector notations: – Just a name, an abstraction that refers to something physical r – Or alternatively as a vector in an M-dimensional complex-valued vector space • Just as calculus provides the mathematical basis for Classical Mechanics, the mathematical basis for QM is linear algebra – Vectors, matrices, eigenvalues, rotations, etc… are key concepts r r ! – Unit vectors are predefined in physical r r terms r1 (t) = r " e1 – Components are projections onto unit ! r r r r vectors e !e = 0 e !e =1 1 2 1 – Unit vectors are orthonormal 1 ! 3. Column vector: – Unit vectors are implied – We will refer to this abstract vector space as `Hilbert Space’ or `state space’ – Any vector in this space corresponds to a possible quantum-mechanical state. The number of such quantum states is uncountable infinity r 2. Unit vectors: r (t) = r1 (t)e1 + r2 (t)e2 + r3 (t)e3 Consider a system with M distinct available states – The 2M real numbers can be viewed as a vector in an 2M-dimensional real-valued vector space r r (t) 1. Vector notation: " r1 (t) % r $ ' r (t) = $ r2 (t)' $ ' # r3 (t)& `Dirac notation’: – Just new symbols for ! same concepts `ket’ r r (t ) # ! (t ) , " (t ) , % (t ) ,... `bra’ rT r (t ) # ! (t ) , " (t ) , % (t ) ,... r e j # j , n , an , r , p , n, m , En , En , m ,... r r r r r t ) = r1 (t )e1 + r2 (t )e2 + r3 (t )e3 # ! (t ) = c1 (t ) 1 + c2 (t ) 2 + c3 (t ) 3 r r r r a $ b = aT b = a b `inner product’ r r rj (t ) = e j $ r (t ) # c j (t ) = j ! (t ) Added catch since QM vectors are complex • Norm of a vector: – Transpose operation replaced by ‘Hermitian conjugation’ or ‘dagger’ operation ba = ab " – a.k.a. magnitude, length r r r = r !r r r = rTr ! = !! – ‘†’ is transpose plus complex conjugation ! r rT r T = (r ) " # = ( # ) † • To compute the norm in terms of the components along a set of orthogonal unit vectors: • Projectors and Closure relations: r r r r r ! r = r1e1 + r2e2 + K + rM eM r r r r r r r r r = e1 (e1 " r ) + e2 (e2 " r ) + K + eM (eM " r ) – Insert the identity ) ) = ) 1) (M % = ) && ! j j ## ) ' j =1 $ " = c1 1 + c1 1 + K + c M M ! = 1 1" + 2 2 " +K+ M M " M =! ) j j) = (1 1 + 2 2 +K+ M M )" – This proves the `closure relation’: ! j =1 M "j j=1 The summation is over a complete set of unit vectors that spans any Hilbert sub-space is equal to the identity operator in that sub-space ! •The entire Hilbert space is a trivial sub-space M M j =1 j =1 = ! c "j c j = ! c j j =1 Old notation: 2 r r r " r = r12 + r22 + K = ! rj2 j Avoid being confused by implied meanings of various symbols Summary • There are ‘ket’s and ‘bra’s: • To avoid confusion, keep in mind that ‘| !’ indicates a Hilbert-space vector, the ‘"’ in |" !’ is just a label – We could call it anything • |" ! , |# ! , |$ ! , |3 ! , |Alice ! – We just need to clearly define our labels • “let |"(t) ! be the state of our system at time t.” • “let |x ! be the state in which the particle lies at position x.” – Here x is a placeholder which could take on any numerical value. I.e. defining the state |x ! as above actually defines an infinite set of vectors, one for each point on the real axis. – This is exactly how the symbol ‘x ’ is used when you say `f(x) = cos(x)’ • ‘let |j ! be the state in which our system is in the jth quantized energy level. – Here j is a placeholder for an arbitrary integer – ket: |" ! • A ket is a vector in an M dimensional Hilbert space, where M is the number of distinct physical states of a system – bra: %"| • A bra is a transposed, conjugated ket • Put a bra and a ket together to get a cnumber – %"|#! := a c-number • c-number := complex number • Unit vectors: – An M dimensional Hilbert space is spanned by M orthonormal unit vectors – {|j!}={|1!, |2!, |3!, …, |M! } ( { } = ‘the set of’ ) – %j|k! = &jk ( &jk is ‘Kronecker delta function’) » 1 if j=k » 0 else – Closure relation: M "j j=1 ! j =1