Prog. Theor. Phys. Vol. 21 No. 6 (1959) pp. 902-918
On the Space-Time Formulation of Non-Relativistic Quantum Mechanics
Department of Physics, University of Osaka Prefecture, Sakai
(Received February 23, 1959)
For the case of the quantization of the usual non-relativistic
classical Lagrangian function quadratic in the velocity the validity
is demonstrated of the non-canonical space-time formulation of quantum
mechanics proposed recently by the author, which aims to evaluate,
without appealing to the Schrodinger equation, the transformation
function K(x, t′′; y, t′) in the space
representation on the basis of the composition rule
K(x, t′′; y, t′) = \intK(x, t′′;
z, t)dzK(z, t; y, t′) (1)
coupled with the supposition that it is approximated to zeroth order
in the quantum of action h by the so-called semi-classical kernel
Kc(x, t′′; y, t′) = [(i/h)∂2 S /
∂x ∂y]1/2 exp [(i / \hslash) S (x, t′
′; y, t′] (2)
written in terms of the classical action
S(x, t′′; y, t′) alone.
In the first place the action function corresponding to the above Lagrangian is expanded in power of the interval of time T = t′′ - t′. Then the deviation of the semi-classical kernel (2) from the unitary transformation function is shown to be of the third order in T, and the corresponding correction term is evaluated by solving the integral equation (1). It is also shown that the semi-classical kernel is unitary for a free motion of a particle with its mass being a function in the space coordinate.
DOI : 10.1143/PTP.21.902
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Citing Article(s) :
Progress of Theoretical Physics Vol. 38 No. 1 (1967) pp. 1-8
Hamilton's Principle from a Quantal Point of View
Progress of Theoretical Physics Vol. 48 No. 4 (1972) pp. 1395-1407
On the Consistency between Lagrangian and Hamiltonian Formalisms in Quantum Mechanics. III
Toshiei Kimura, Teruya Ohtani and Reiji Sugano
Progress of Theoretical Physics Vol. 50 No. 1 (1973) pp. 277-289
Quantization for Non-Linear Lagrangian