Quantum Mechanics for Scientists and Engineers. David Miller

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1 Quantum Mechanics for Scientists and Engineers David Miller

2 Unitary and Hermitian operators

3 Unitary and Hermitian operators Using unitary operators

4 Unitary operators to change representations of vectors Suppose that we have a vector (function) that is represented when expressed as an expansion on the functions n as the mathematical column vector These numbers c 1, c 2, c 3, are the projections of f old on the orthogonal coordinate axes in the vector space labeled with,, f old f old c c c 1 2 3

5 Unitary operators to change representations of vectors Suppose we want to represent this vector on a new set of orthogonal axes which we will label 1, 2, 3 Changing the axes which is equivalent to changing the basis set of functions does not change the vector we are representing but it does change the column of numbers used to represent the vector

6 Unitary operators to change representations of vectors For example, suppose the original vector was actually the first basis vector in the old basis 1 Then in this new representation the elements in the column of numbers would be the projections of this vector on the various new coordinate axes each of which is simply m So under this coordinate transformation or change of basis f old

7 Unitary operators to change representations of vectors Writing similar transformations for each basis vector we get the correct transformation if we define a matrix u11 u12 u13 ˆ u21 u22 u23 U u31 u32 u33 where u ij i j and we define our new column of numbers f Uˆ f new old n f new

8 Unitary operators to change representations of vectors Note incidentally that here f and f are the same vector in the vector space old new Only the representation the coordinate axes and, consequently the column of numbers that have changed not the vector itself

9 Unitary operators to change representations of vectors Now we can prove that Uˆ is unitary Writing the matrix multiplication in its sum form ˆ ˆ UU ij umiu m mj m i m j i m m j m m i m m j ˆ i I j i j ij m so UU ˆ ˆ Iˆ hence Uˆ is unitary since its Hermitian transpose is therefore its inverse

10 Unitary operators to change representations of vectors Hence any change in basis can be implemented with a unitary operator We can also say that any such change in representation to a new orthonormal basis is a unitary transform Note also, incidentally, that ˆˆ ˆ ˆ ˆ UU U U I Iˆ so the mathematical order of this multiplication makes no difference

11 Unitary operators to change representations of operators Consider a number such as gaf ˆ where vectors f and g and operator  are arbitrary This result should not depend on the coordinate system so the result in an old coordinate system g ˆ old Aold fold should be the same in a new coordinate system that is, we should have g ˆ ˆ new Anew fnew gold Aold fold Note the subscripts new and old refer to representations not the vectors (or operators) themselves which are not changed by change of representation Only the numbers that represent them are changed

12 Unitary operators to change representations of operators With unitary Uˆ operator to go from old to new systems we can write g Aˆ f g Aˆ f Since we believe also that then we identify or since new new new new new new U ˆ g ˆ ˆ old Anew U fold ˆ ˆ g Aˆ f g Aˆ f new new new old old old ˆ Aold U AnewU ˆ ˆ ˆ old new new UA ˆ Uˆ UU ˆˆ A UU ˆˆ A ˆ ˆ ˆ ˆ old new old g U A U f then Aˆ new UA ˆˆ Uˆ old

13 Unitary operators that change the state vector For example, if the quantum mechanical state is expanded on the basis n to give 2 then an 1 n and if the particle is to be conserved then this sum is retained as the quantum mechanical system evolves in time But this is just the square of the vector length Hence a unitary operator, which conserves length describes changes that conserve the particle an n n

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15 Unitary and Hermitian operators Hermitian operators

16 Hermitian operators A Hermitian operator is equal to its own Hermitian adjoint Mˆ Mˆ Equivalently it is self-adjoint

17 Hermitian operators In matrix terms, with M11 M12 M13 ˆ M21 M22 M23 M M 31 M 32 M 33 then M12 M22 M31 M13 M23 M33 so the Hermiticity implies M ij M ji for all i and j so, also the diagonal elements of a Hermitian operator must be real Mˆ M M M

18 Hermitian operators To understand Hermiticity in the most general sense consider gmˆ f for arbitrary f and g and some operator ˆM We examine gmˆ f Since this is just a number a 1 x 1 matrix it is also true that gmˆ f gmˆ f

19 Hermitian operators We can also analyze using the rule for Hermitian adjoints of products So ˆ gm f gmˆ f Hence, if ˆM is Hermitian, with therefore then gmˆ f f Mg ˆ ˆ f M gmˆ f ˆ ˆ ˆ ˆ AB B A ˆM f g f M g even if f and g are not orthogonal This is the most general statement of Hermiticity g Mˆ Mˆ ˆ

20 Hermitian operators f x In integral form, for functions and g x the statement gmˆ f f Mg ˆ can be written ˆ g x Mf xdx f xmg ˆ xdx We can rewrite the right hand side using ab g x Mf ˆ x dx f x Mg ˆ x dx and a simple rearrangement leads to g x Mf ˆ x dx Mg ˆ x f x dx which is a common statement of Hermiticity in integral form a b

21 Bra-ket and integral notations Note that in the bra-ket notation the operator can also be considered to operate to the left g Aˆ is just as meaningful a statement as  f and we can group the bra-ket multiplications as we wish g A ˆ f g A ˆ f g A ˆ f Conventional operators in the notation used in integration such as a differential operator, d/dx do not have any meaning operating to the left so Hermiticity in this notation is the less elegant form g x Mf ˆ x dx Mg ˆ x f x dx

22 Reality of eigenvalues n Suppose is a normalized eigenvector of the Hermitian operator ˆM with eigenvalue n Then, by definition Mˆ n n n Therefore Mˆ n n n n n n But from the Hermiticity of ˆM we know Mˆ Mˆ and hence must be real n n n n n n

23 Orthogonality of eigenfunctions for different eigenvalues Trivially By associativity Using Using Hermiticity Using m and are real numbers Rearranging ˆ ˆ ˆ ˆ AB B A Mˆ Mˆ n n n Mˆ Mˆ Mˆ Mˆ 0 m n m n 0 m n m n 0 Mˆ ˆ m n m M n ˆ M 0 Mˆ ˆ m n m M n 0 m m n m n n 0 m m n n m n But and are different, so i.e., orthogonality m n n 0 m n m n 0 m n

24 Degeneracy It is quite possible and common in symmetric problems to have more than one eigenfunction associated with a given eigenvalue This situation is known as degeneracy It is provable that the number of such degenerate solutions for a given finite eigenvalue is itself finite

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26 Unitary and Hermitian operators Matrix form of derivative operators

27 Matrix form of derivative operators Returning to our original discussion of functions as vectors we can postulate a form for the differential operator where we presume we can take the limit as x d x dx x x 0 x

28 Matrix form of derivative operators If we multiply the column vector whose elements are the values of the function then 1 1 f xi x i x i df 0 0 f x f x x 2 x 2x f xi 2x dx x i f x 2 0 i x f xi x f x i df 2 x 2 x f xi 2 x 2 x dx xi x where we are taking the limit as x 0 Hence we have a way of representing a derivative as a matrix

29 Matrix form of derivative operators Note this matrix is antisymmetric in reflection about the diagonal and it is not Hermitian Indeed somewhat surprisingly d/dx is not Hermitian By similar arguments, though d 2 /dx 2 gives a symmetric matrix and is Hermitian d dx x 2 x x 2 x

30 Matrix corresponding to multiplying by a function We can formally operate on the function f x by multiplying it by the function V x to generate another function g x V x f x Since V x is performing the role of an operator we can if we wish represent it as a (diagonal) matrix whose diagonal elements are the values of the function at each of the different points If Vx is real then its matrix is Hermitian as required for Ĥ

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