Unified Description of Matrix Mechanics and Wave Mechanics on Hydrogen Atom

Topics: Quantum mechanics, Matrix, Schrödinger equation Pages: 38 (8730 words) Published: July 22, 2013
Unified Description of Matrix Mechanics and Wave Mechanics on Hydrogen Atom

arXiv:1201.0136v5 [math-ph] 25 Mar 2013

Yongqin Wang1 , Lifeng Kang2
1

Department of Physics, Nanjing University, Nanjing 210008, China e-mail:yhwnju@hotmail.com

2

Faculty of Science, National University of Singapore, Singapore 117543

Abstract A new mathematical method is established to represent the operator, wave functions and square matrix in the same representation. We can obtain the specific square matrices corresponding to the angular momentum and RungeLenz vector operators with invoking assistance from the operator relations and the orthonormal wave functions. Furthermore, the first-order differential equations will be given to deduce the specific wave functions without using the solution of the second order Schr¨dinger equation. As a result, we will o unify the descriptions of the matrix mechanics and the wave mechanics on hydrogen atom. By using matrix transformations, we will also deduce the specific matrix representations of the operators in the SO(4,2) group. Keywords: Matrix Operator Wave function Hydrogen atom Quantum mechanics 1. Introduction In 1925, based on Niels Bohr’s correspondence principle, Werner Heisenberg represented the spatial coordinate q and the momentum p by the following form [1] q = [q(nm)e2πiν(nm)t ], p = [p(nm)e2πiν(nm)t ] (1)

Preprint submitted to Elsevier

March 26, 2013

They abandoned the representation (1) in favor of the shorter notation q = q(nm) p = p(nm)

Max Born and Pascual Jordan then wrote q substituted for q(nm) as a matrix [2]   0 q(01) 0 0 0 ···  q(10) 0 q(12) 0 0 ···     0 q(21) 0 q(23) 0 · · ·  ··· ··· ··· ··· ··· ···

The founders of matrix mechanics tried to describe the mechanics quantum by the square matrix. They had not been successful because the source of the matrix could not be explained. In modern quantum mechanics the mechanical quantities was described by the operator. However, the operator was studied in isolation without being related to the wave functions so that the square matrix in quantum mechanics looked very mysterious. In fact, the square matrix is derived from the superposition coefficient. To address this issue, we offer the following new approaches. In linear algebra, the product of row matrix and the same order column matrix is equal to a polynomial. For example A1 A2 B1 B2 = A1 B1 + A2 B2

Now that equality holds from left to right, the equality should hold from right to left B1 (2) A1 B1 + A2 B2 = A1 A2 B2 So a polynomial can be expanded to the product of row matrix and the same order column matrix. Therefore I, The following expressions are equivalent aψ1 + bψ2 cψ1 + dψ2 and ψ1 ψ2 = ψ1 ψ2 a c b d

a c b d

=

aψ1 + bψ2 cψ1 + dψ2

2

II, If

ˆ Aψ1 = aψ1 + bψ2 ˆ Aψ2 = cψ1 + dψ2

(3)

then ˆ A ψ1 ψ2 III, If = aψ1 + bψ2 cψ1 + dψ2 = ψ1 ψ2 a c b d (4)

ˆ Bψ1 = eψ1 + f ψ2 ˆ Bψ2 = gψ1 + hψ2 ˆ B ψ1 ψ2 = ψ1 ψ2 e g f h

(5)

then

(6)

Thus, from (4) and (6), ˆˆ AB ψ1 ψ2 ˆ = A ψ1 ψ2 e g f h = ψ1 ψ2 a c b d e g f h (7)

In fact, we are also able to get from (3) and (5) ˆˆ AB ψ1 ψ2 = ˆ ˆ eAψ1 + f Aψ2 ˆ ˆ g Aψ1 + hAψ2 = e f g h ˆ Aψ1 ˆ Aψ2 = e f g h a b c d ψ1 ψ2

The expression (7) is clearer than the above expression, so (3) is represented as (4) and we adopt (7) in this article. From (122) and (124),  √ 3  L+ Y1−1 = cos θ = 2 Y10  ˆ 2π  √ 3 (8) ˆ L+ Y10 = − 4π sin θeiϕ = 2 Y11    ˆ L+ Y11 = 0 (8) can be represented as ˆ L+   0 0 0  1 0 0  0 1 0

Y1−1 Y10 Y11

=

Y1−1 Y10 Y11

(9)

3

The square matrix in (9) is just one of the results of matrix mechanics on the angular moment. In fact, we can convert the operator relations (16)-(18) into the square matrix relations with invoking assistance from (20) and (21). The specific values of the eigenvalues in (20) and the matrix elements in (21) can be deduced from those square matrix relations. Furthermore, the first order differential equations are got...
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