- #1
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In Dirac's "The Principles of Quantum Mechanics", in chapter V on the equations of motion Dirac proceeds with a line of reasoning that is something along the following lines (I've modified it a bit to coincide with what's taught in the course I'm taking)
1. We assume that the motion throughout the time that the system is undisturbed can be predicted given the initial state of the system \left|\psi{t_0}\right\rangle, up to a complex scalar.
2. Owing to the principle of superposition, the scalar product between two states is constant in time, that is to say, \frac{d}{dt}\left\langle \phi\left(t\right)|\psi\left(t\right)\right\rangle=0
3. If we assume the time derivative of the state of the system is described by a linear operator, then this linear operator must be anti-hermitic, we denote it -iA
4. If we assume that A is the sum of a function of the momentum operator and a function of the position operator, then in order to satisfy the correspondence principle for Hamilton's equations, we must define H=\hbar A and come to the conclusion that SE holds.
I'm fairly sure of steps 1, 3 and 4, but step 2 has me really confused.
In today's lecture, the professor wasn't quite able to justify it on the spot, he said he'd have an answer for us eventually, but he's a bit flaky on those kinds of promises, so I'd like to see if anyone here could help me.
Dirac's explanation is as follows:
"The requirement that the state a tone time determines
the state a t another time means that \left|\psi{t_0}\right\rangle determines \left|\psi{t}\right\rangle except for a numerical factor. The principIe of superposition applies to these
states of motion throughout the time during which the system is
undisturbed, and means that if we take a superposition relation
holding for certain states at time t_0 and giving rise to a linear equation
between the corresponding kets, e.g. the equation
\left|R{t_0}\right\rangle=c_1 \left|A{t_0}\right\rangle + c_2 \left|B{t_0}\right\rangle
the same superposition relation must hold between the states of
motion throughout the thne during which the system is undisturbed
and must lead to the same equation between the kets corresponding to these states at any time t (in the undisturbed time interval), i.e. the equation
\left|R{t}\right\rangle=c_1 \left|A{t}\right\rangle + c_2 \left|B{t}\right\rangle
provided the arbitrary numerical factors By which these kets may be multiplied are suitably chosen. It follows that the \left|P{t}\right\rangle's are linear
functions of the \left|P{t_0}\right\rangle's and each \left|P{t_0}\right\rangle is the result of some linear
operator applied to \left|P{t_0}\right\rangle"
From there the argument proceeds to say that we introduce a physical assumption that suitable constants can always be found so that the length of the state vector is constant in time.
In essence, the argument reads that the time-evolved vector is the result of some unitary operator acting on the vector at t_0
If I can accept this claim, then I'm done, since the conservation of the inner-product with time follows immediately.
To be quite frank, I lost Dirac's argument at the point he introduced R,A,B and would be very glad if someone could help me over that hurdle.
-Anatoli
1. We assume that the motion throughout the time that the system is undisturbed can be predicted given the initial state of the system \left|\psi{t_0}\right\rangle, up to a complex scalar.
2. Owing to the principle of superposition, the scalar product between two states is constant in time, that is to say, \frac{d}{dt}\left\langle \phi\left(t\right)|\psi\left(t\right)\right\rangle=0
3. If we assume the time derivative of the state of the system is described by a linear operator, then this linear operator must be anti-hermitic, we denote it -iA
4. If we assume that A is the sum of a function of the momentum operator and a function of the position operator, then in order to satisfy the correspondence principle for Hamilton's equations, we must define H=\hbar A and come to the conclusion that SE holds.
I'm fairly sure of steps 1, 3 and 4, but step 2 has me really confused.
In today's lecture, the professor wasn't quite able to justify it on the spot, he said he'd have an answer for us eventually, but he's a bit flaky on those kinds of promises, so I'd like to see if anyone here could help me.
Dirac's explanation is as follows:
"The requirement that the state a tone time determines
the state a t another time means that \left|\psi{t_0}\right\rangle determines \left|\psi{t}\right\rangle except for a numerical factor. The principIe of superposition applies to these
states of motion throughout the time during which the system is
undisturbed, and means that if we take a superposition relation
holding for certain states at time t_0 and giving rise to a linear equation
between the corresponding kets, e.g. the equation
\left|R{t_0}\right\rangle=c_1 \left|A{t_0}\right\rangle + c_2 \left|B{t_0}\right\rangle
the same superposition relation must hold between the states of
motion throughout the thne during which the system is undisturbed
and must lead to the same equation between the kets corresponding to these states at any time t (in the undisturbed time interval), i.e. the equation
\left|R{t}\right\rangle=c_1 \left|A{t}\right\rangle + c_2 \left|B{t}\right\rangle
provided the arbitrary numerical factors By which these kets may be multiplied are suitably chosen. It follows that the \left|P{t}\right\rangle's are linear
functions of the \left|P{t_0}\right\rangle's and each \left|P{t_0}\right\rangle is the result of some linear
operator applied to \left|P{t_0}\right\rangle"
From there the argument proceeds to say that we introduce a physical assumption that suitable constants can always be found so that the length of the state vector is constant in time.
In essence, the argument reads that the time-evolved vector is the result of some unitary operator acting on the vector at t_0
If I can accept this claim, then I'm done, since the conservation of the inner-product with time follows immediately.
To be quite frank, I lost Dirac's argument at the point he introduced R,A,B and would be very glad if someone could help me over that hurdle.
-Anatoli