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)⟩=iℏd|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]=AB−BA=0, and incompactible if [A,B]≠0
Schwarz Inequality
Let an and bn be two sets of sequences. The Schwarz inequality states that (n∑ia2i)(n∑ib2i)≥(n∑iaibi)2
Proof
Let S(n) be the statement
(n∑ia2i)(n∑ib2i)≥(n∑iaibi)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.
(k∑ia2i)(k∑ib2i)≥(k∑iaibi)2.
When n=k+1,
R.S.=(k+1∑iaibi)2=(k∑iaibi+ak+1bk+1)2=(k∑iaibi)2+2ak+1bk+1(k∑ib2i)(∑ka2i)+a2k+1b2k+1
L.S.=(k+1∑ia2i)(k+1∑ib2i)=(a2k+1+k∑ia2i)(b2k+1+k∑ib2i)=a2k+1b2k+1+a2k+1k∑ib2i+∑ka2ib2k+1+(k∑ia2i)(k∑ib2i)≥a2k+1b2k+1+a2k+1k∑ib2i+b2k+1∑ka2i+(k∑iaibi)2,
by S(k). Substitue right hand side into the inequality,
L.S.≥a2k+1b2k+1+a2k+1k∑ib2i+b2k+1∑ka2i+(k∑iaibi)2=(k+1∑iaibi)2−2ak+1bk+1(k∑ib2i)(∑ka2i)+a2k+1k∑ib2i+b2k+1∑ka2i.
Examining the last four terms, we have
2ak+1bk+1(k∑ib2i)(∑ka2i)+a2k+1k∑ib2i+b2k+1∑ka2i=(ak+1−2ak+1bk+1∑kiaibi∑kib2i)k∑ib2i+b2k+1k∑ia2i=(ak+1−bk+1∑kiaibi∑ib2i)2(k∑ib2i)−(k∑ib2i)b2k+1(∑kiaibi∑ib2i)+b2k+1k∑ia2i=(ak+1−bk+1∑kiaibi∑ib2i)2(k∑ib2i)+b2k+1(−(∑kiaibi)2+∑kia2i∑kib2i∑kib2i),
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+1∑ia2i)(k+1∑ib2i)≥(k+1∑iaibi)2−2ak+1bk+1(k∑ib2i)(∑ka2i)+a2k+1k∑ib2i+b2k+1∑ka2i≥(k+1∑iaibi)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=∑niai→ei and →b=∑nibi→ei 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|f⟩⟨g|g⟩≥|⟨f|g⟩|2.
Uncertainty Principle
Proof taken from Griffiths. The standard derivation of an observable A is given by σ2A+=⟨(ˆA−⟨A⟩)2⟩=⟨Ψ|(ˆA−⟨A⟩)2|Ψ⟩=⟨(ˆA−⟨A⟩)Ψ|(ˆA−⟨A⟩)Ψ⟩=⟨f|f⟩, where f≡(ˆA−⟨A⟩)Ψ. Similarly, for another observable B, σ2B=⟨g|g⟩, where g≡(ˆB−⟨B⟩)Ψ. By Schwarz inequality, σ2Aσ2B=⟨f|f⟩⟨g|g⟩≥|⟨f|g⟩|2. For any complex number z, |z|2=[Re(z)]2+[Im(z)]2≥[Im(z)]2=[12i(z−z∗)]2. This is also true for ⟨f|g⟩, so σ2Aσ2B≥|⟨f|g⟩|2≥(12i[⟨f|g⟩−⟨g|f⟩])2. Compute the first term on the right hand side inside the bracket, we have ⟨f|g⟩=⟨(ˆA−⟨A⟩)Ψ|(ˆB−⟨B⟩)Ψ⟩=⟨Ψ|(ˆA−⟨A⟩)(ˆB−⟨B⟩)|Ψ⟩=⟨Ψ|(ˆAˆB−ˆA⟨B⟩−ˆB⟨A⟩+⟨A⟩⟨B⟩)|Ψ⟩=⟨ˆAˆB⟩−⟨B⟩⟨A⟩−⟨A⟩⟨B⟩+⟨A⟩⟨B⟩=⟨ˆAˆB⟩−⟨A⟩⟨B⟩, where ⟨A⟩ and ⟨B⟩ are real numbers so they can be taken out from ⟨|Ψ⟩ and they commute. Similarly, we have ⟨g|f⟩=⟨ˆBˆA⟩−⟨A⟩⟨B⟩. Thus, ⟨f|g⟩−⟨g|f⟩=⟨ˆAˆB⟩−⟨ˆBˆA⟩=⟨[ˆA,ˆB]⟩. Substitute back, we have σ2Aσ2B≥(12i⟨[ˆA,ˆB]⟩)2. This is called the uncertainty principle.
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