Normalization

A mathematical feature of the Schrödinger equation is that, if ψ is a solution, then so is Nψ, where Nis any constant. This feature is confirmed by noting that ψ occurs in every term in eqn 8.13, so any constant factor can be cancelled. This freedom to vary the wavefunction by a constant factor means that it is always possible to find a normalization constant, N, such that the proportionality of the Born interpretation becomes an equality.

We find the normalization constant by noting that, for a normalized wavefunction Nψ, the probability that a particle is in the region dx is equal to (Nψ*) (Nψ) dx (we are taking N to be real). Furthermore, the sum over all space of these individual probabilities must be 1 (the probability of the particle being somewhere is 1). Expressed mathematically, the latter requirement is

Almost all wavefunctions go to zero at sufficiently great distances so there is rarely any difficulty with the evaluation of this integral, and wavefunctions for which the integral in eqn 8.15 exists (in the sense of having a finite value) are said to be ‘square integrable’. It follows that

Therefore, by evaluating the integral, we can find the value of N and hence ‘normalize’ the wavefunction. From now on, unless we state otherwise, we always use wave functions that have been normalized to 1; that is, from now on we assume that ψ already includes a factor that ensures that (in one dimension)

In three dimensions, the wavefunction is normalized if

or, more succinctly, if

where dτ = dxdydz and the limits of this definite integral are not written explicitly: in all such integrals, the integration is over all the space accessible to the particle. For systems with spherical symmetry it is best to work in spherical polar coordinates r, θ, and φ(Fig. 8.22): x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. The volume element in spherical polar coordinates is dτ = r² sinθ dr dθ dφ. To cover all space, the radius r ranges from 0 to ∞, the colatitude, θ, ranges from 0 to π, and the azimuth, φ, ranges from 0 to 2π (Fig. 8.23), so the explicit form of eqn 6.17c is

Fig. 8.22 The spherical polar coordinates used for discussing systems with spherical symmetry.

Fig. 8.23 The surface of a sphere is covered by allowing θ to range from 0 to π, and then sweeping that arc around a complete circle by allowing φ to range from 0 to 2π.

اخر الاخبار

اخبار العتبة العباسية المقدسة

لتعزيز الوعي الديني… قسم الشؤون الدينية يقدم دوراته الفقهية لمنتسبي العتبة العباسية المقدسة الجدد

لأكثر من 150 طالبًا.. قسم الشؤون الفكرية ينظم مسابقة معرفية في بغداد

جامعة الكفيل تكرم المشاركين في فعاليات المؤتمر الطلابي السادس لمشاريع تخرج الطلبة

إعلان أسماء الفائزين في مشاريع التخرج ضمن اليوم الثاني للمؤتمر الطلابي السادس

المزيد

الآخبار الصحية

سرّ بسيط لصحة القلب.. أطعمة تخفض الكوليسترول خلال أسابيع

سبب خفي لإعتام عدسة العين ؟

عوامل خفية تسرّع شيخوخة الدماغ

أسباب الرغبة الشديدة في تناول الحلويات والأطعمة النشوية

المزيد

الآخبار التكنلوجية

كندا تصنع كاسحة جليد من جيل جديد

رواد "أرتيميس 2" يكشفون تفاصيل حياتهم في الفضاء

حفريات صينية تكشف كائنات بحرية عاشت قبل 546 مليون عام

روسيا تطلق صاروخ "سويوز-2.1 إيه" من قاعدة بليسيتسك الفضائية

المزيد

مواضيع ذات صلة

Units

Names of quantities

Working galvanic cells

Electrolysis

Experimental techniques

The rate laws

The electric potential at the interface

The structure of the interface

Catalytic activity at surfaces

Mechanisms of heterogeneous catalysis

Mobility on surfaces

The rate of desorption