Divergence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>〈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> <mml:mo>〉</mml:mo> </mml:math> in discontinuous potential wells
Abstract
Abstract The surprising divergence of the expectation value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">〈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">〉</mml:mo> </mml:math> for the square well potential is known. Here, we prove and demonstrate the divergence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">〈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">〉</mml:mo> </mml:math> in potential wells which have a finite jump discontinuity. Apart from the square well, two-piece half-potential wells are such examples. These half-potential wells can be expressed as V ( x ) = − U ( x ) Θ ( x ), where Θ( x ) is the Heaviside step function. U ( x ) are continuous and differentiable functions with a minimum at x = 0 and which may or may not vanish as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>x</mml:mi> <mml:mo>∼</mml:mo> <mml:mo>∞</mml:mo> </mml:math> .