Show that a) |hx, yi ≤ ||x|| ||y

Zielona G´ora, 17 October 2015

Quantum Mechanics CLASS

Section 2 Problem 1)

Let hx, yi denote the scalar product in a linear space X , przy czym ||x||² = hx, xi. Show that

a) |hx, yi ≤ ||x|| ||y||, (the Schwartz inequality);

b) ||x + y||² + ||x − y||² = 2(||x||² + ||y||²), (the rectangular equivalence);

c) hx, yi = 0 ⇔ ||x + y||² = ||x||²+ ||y||², (the Pitagoras theorem).

Problem 2)

Show that eigenvalues of the operator ˆF = ( ˆL⁺)ⁿ( ˆL)ⁿ s¸a are not negative.

Problem 3)

Show that if ˆU is a unitary operator then

a) the eigenvalues of ˆU are complex numbers with the modulus 1;

b) operators ˆA and ˆAU = ˆU ˆA ˆU⁺ have the same eigenvalues;

c) if operator ˆA is self-adjoint then ˆA_U = ˆU ˆA ˆU⁺ is self-adjoint as well.

Problem 4)

The commutator of the operators ˆx, ˆp is equal [ˆp, ˆx] = −i¯h , Find

a) [ˆp, ˆx²];

b) [ˆp, ˆxⁿ];

c) [ˆp, f (ˆx)]; assume that f (x) can be expanded in the Taylor series;

d)assuming

ˆ

x ψ(x) = x ψ(x), find

ˆ p ψ(x) . Problem 5)

Suppose ˆK and ˆF do not commutate, [ ˆK, ˆF ] = i ˆM 6= 0; ˆK, ˆF , ˆM are self- adjoint. Prove the uncertainty principle for ˆK and ˆF .

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Hint. Define a mean value (expectation value) for the above operators in a given state ψ as well as standard deviations ∆ ˆF , ∆ ˆK. Prove

∆ ˆF ∆ ˆK ≥ 1

2|h ˆM i| ,

where h ˆM i is the mean value of ˆM . To prove the above inequality consider the operator ˆF +iλ ˆK, where λ ∈ R and use the fact (ψ, ( ˆF −iλ ˆK)( ˆF +iλ ˆK)ψ) ≥ 0.

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