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Date: 13-7-2016
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Date: 30-8-2016
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Date: 13-7-2016
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Combined Potential
A particle of mass m is confined to x > 0 in one dimension by the potential
(i)
where V0 and b are constants. Assuming there is a bound state, derive the exact ground state energy.
SOLUTION
Let the dimensionless distance be y = x/b. The kinetic energy has the scale factor Eb = h2/2mb2. In terms of these variables we write Schrodinger’s equation as
(1)
Our experience with the hydrogen atom, in one or three dimensions, is that potentials which are combinations of y-1 and y-2 are solved by exponentials e-αy times a polynomial in y. The polynomial is required to prevent the particle from getting too close to the origin where there is a large repulsive potential from the y-2 term. Since we do not yet know which power of y to use in a polynomial, we try
(2)
where s and α need to be found, while A is a normalization constant. This form is inserted into the Hamiltonian. First we present the second derivative from the kinetic energy and then the entire Hamiltonian:
(3)
(4)
We equate terms of like powers of y:
(5)
(6)
(7)
The last equation defines s. The middle equation defines α once is known. The top equation gives the eigenvalue:
(8)
(9)
(10)
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دراسة يابانية لتقليل مخاطر أمراض المواليد منخفضي الوزن
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اكتشاف أكبر مرجان في العالم قبالة سواحل جزر سليمان
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اتحاد كليات الطب الملكية البريطانية يشيد بالمستوى العلمي لطلبة جامعة العميد وبيئتها التعليمية
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