Postulates of Quantum Mechanics

Postulates of Quantum Mechanics

  • All properties of a quantum system can be found with its state vector |ψ, which belongs to the complex Hilbert space
  • Every observable physical property has its corresponding Hermittian operator. The eigenvalues of its operator are the possible outcome values when measuring the physical property M|λn=λn|λn. The set of eigenvectors span the Hilbert space.
  • The probability of measuring M to have the value λn of the quantum system in state vector |ψ is given by P=|λ|ψ|2
  • If the result after measuring M is λn, then the state vector of the system immediately after the measurment is |λn
  • The evolution of the state of the quantum system is governed by the Schrodinger equation H|ψ(t)=id|ψ(t)dt

Compactibility

An operator A is said to be compactible to an operator B if A commutes with B, i.e. [A,B]=ABBA=0, and incompactible if [A,B]0

Schwarz Inequality

Let an and bn be two sets of sequences. The Schwarz inequality states that (nia2i)(nib2i)(niaibi)2

Proof
Let S(n) be the statement (nia2i)(nib2i)(niaibi)2. When n=1, L.S.=a21+b21 while R.S.=(a1b1)2, so left hand side equals right hand side. S(1) is true. Assume S(k) is true, i.e. (kia2i)(kib2i)(kiaibi)2. When n=k+1, R.S.=(k+1iaibi)2=(kiaibi+ak+1bk+1)2=(kiaibi)2+2ak+1bk+1(kib2i)(ka2i)+a2k+1b2k+1 L.S.=(k+1ia2i)(k+1ib2i)=(a2k+1+kia2i)(b2k+1+kib2i)=a2k+1b2k+1+a2k+1kib2i+ka2ib2k+1+(kia2i)(kib2i)a2k+1b2k+1+a2k+1kib2i+b2k+1ka2i+(kiaibi)2, by S(k). Substitue right hand side into the inequality, L.S.a2k+1b2k+1+a2k+1kib2i+b2k+1ka2i+(kiaibi)2=(k+1iaibi)22ak+1bk+1(kib2i)(ka2i)+a2k+1kib2i+b2k+1ka2i. Examining the last four terms, we have 2ak+1bk+1(kib2i)(ka2i)+a2k+1kib2i+b2k+1ka2i=(ak+12ak+1bk+1kiaibikib2i)kib2i+b2k+1kia2i=(ak+1bk+1kiaibiib2i)2(kib2i)(kib2i)b2k+1(kiaibiib2i)+b2k+1kia2i=(ak+1bk+1kiaibiib2i)2(kib2i)+b2k+1((kiaibi)2+kia2ikib2ikib2i), in which the first term is non-negative as it is a product of two non-negative number, while the second term is also non-negative, by our assumption S(k) is true. Hence, it is a non-negative number. So, we conclude L.S.=(k+1ia2i)(k+1ib2i)(k+1iaibi)22ak+1bk+1(kib2i)(ka2i)+a2k+1kib2i+b2k+1ka2i(k+1iaibi)2. Therefore, S(k+1) is also true. By mathematical induction, S(n) is true for all natural number n.

Notice that if we let a=niaiei and b=nibiei be two vectors of n dimensions and ei is the basis vectors, then (niaibi)2 is the inner products of the two vectos and (kia2i), (kib2i) are the inner product with themselves. Then, in terms of vector representation, the Schwarz Inequality can be written as f|fg|g|f|g|2.

Uncertainty Principle

Proof taken from Griffiths. The standard derivation of an observable A is given by σ2A+=(ˆAA)2=Ψ|(ˆAA)2|Ψ=(ˆAA)Ψ|(ˆAA)Ψ=f|f, where f(ˆAA)Ψ. Similarly, for another observable B, σ2B=g|g, where g(ˆBB)Ψ. By Schwarz inequality, σ2Aσ2B=f|fg|g|f|g|2. For any complex number z, |z|2=[Re(z)]2+[Im(z)]2[Im(z)]2=[12i(zz)]2. This is also true for f|g, so σ2Aσ2B|f|g|2(12i[f|gg|f])2. Compute the first term on the right hand side inside the bracket, we have f|g=(ˆAA)Ψ|(ˆBB)Ψ=Ψ|(ˆAA)(ˆBB)|Ψ=Ψ|(ˆAˆBˆABˆBA+AB)|Ψ=ˆAˆBBAAB+AB=ˆAˆBAB, where A and B are real numbers so they can be taken out from |Ψ and they commute. Similarly, we have g|f=ˆBˆAAB. Thus, f|gg|f=ˆAˆBˆBˆA=[ˆA,ˆB]. Substitute back, we have σ2Aσ2B(12i[ˆA,ˆB])2. This is called the uncertainty principle.

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