Show that if ⟨h |Q̂ h⟩=⟨Q̂ h |h⟩for all functions h (in Hilbert space), then ⟨f |Q̂ g⟩=⟨Q̂ f |g⟩

Show that if ⟨h |Q̂ h⟩=⟨Q̂ h |h⟩for all functions h (in...

Key Concepts

Polarization Identity

The polarization identity is a mathematical tool that relates the inner product of two vectors to the inner product of sums and differences of these vectors. It effectively allows the extraction of the bilinear form (or sesquilinear form) from a quadratic form. In the context of proving the equivalence of definitions of Hermitian operators, the polarization identity plays a key role in extending the symmetry of inner products from the diagonal (i.e., self-products) to arbitrary pairs of functions.

Inner Product Properties

The inner product in a Hilbert space is a complex-valued function that satisfies linearity in one argument, conjugate symmetry, and positive-definiteness. These properties are crucial because they allow one to manipulate and relate expressions such as ?f | Qg? and ?Qf | g?. The fact that the inner product behaves in this well-structured manner is what enables proofs involving symmetry and adjoint relationships in operator theory.

Hermitian Operators

A Hermitian operator on a Hilbert space is one that satisfies the property that its inner product with any function and the operator acting on that function is equal to the inner product with the function and the operator acting on it, making the result real. This property ensures that the operator’s eigenvalues are real and that its eigenfunctions form an orthogonal basis, which is central in quantum mechanics and other areas of functional analysis.

Transcript

00:01 Hello everyone we are doing problem 3 .3 of griffiths here it is said for a function h in hill bird space we have the condition h in a product with qh is equals to qh in a product with h and we have to prove that for any function f and g in hilbert space, f in a product with qg is equals to qf in a product with g.

00:42 For this, we will use the condition as suggested in the question.

00:48 We will take h of x equals to f of x plus c into g of x, where c is an arbitrary constant and we will use the condition that this first condition so h in a product with h is equals to f plus cg in a product with q f plus c g so we have just substituted h equals to f plus cg here in this expression now if we expand this we will get f in a product with qf plus c times f in a product with qg.

02:08 We are just expanding the brackets using the multiplication rule plus c star g in a product with qf plus c squared g in a product with qf plus c squared, g in a product with qg so, c star comes here because here it is cg and since it is on the left side of the inner product, it always gets a imaginary sign.

02:51 So that's why the constants, if it is coming out of the integral, it gets a star.

02:57 And here also there was a c star and there was a c coming from the right side.

03:02 So c star c becomes mod of c squared.

03:07 Similarly we will do it for qcaph in a product with h and we will get finally we'll apply the same method we'll get finally to gap f in a product with f plus c q cap f in a product with g plus c star q cap g in a product g in a product f plus c squared, sorry, there is only one c squared here, q star g in a product with g.

04:18 Now we can, we will use the same expression here for f and g as well.

04:26 If we use the same expression in place of h, if we use f, then we see that this expression f in a product with qf is actually equal to q in a q f in a product with f.

04:41 So these two expressions since they are equal using this expression, these two expressions inside the rhs of these two expressions will cancel this one and this one...

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