Normalised Eigenstates

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Exercise:
The eigenstates for a particle in an infinite square potential well are given by psi_nxt Asink_n x e^-iomega_n t Show that for the normalised wave functions i.e. for _^L|psi_nxt|^textrmdx the amplitude has to be A sqrtfracL

Solution:
_^L|psi_nxt|^textrmdx _^L|Asink_n x e^-iomega_n t|^ textrmdx _^L|A|^sin^k_n x |e^-iomega_n t|^ textrmdx |A|^ _^L sin^k_n xtextrmdx where we use the fact that e^iphi is a unit vector in the complex plane. vspacemm Using the identiy sin^phi fracleft-cos phiright we can write the egral as dots frac|A|^ _^L left-cos k_n xright textrmdx frac|A|^ left _^L textrmdx -_^L cos k_n x textrmdx right frac|A|^ L-frac|A|^ k_n sin k_n xBig|_^L frac|A|^L-frac|A|^ k_nsink_n xBig|_^L The sine function has zeros at and also at L by definition of k_n so the second term vanishes. The normalisation condition can thus be written as frac|A|^L Longrightarrow |A|^ fracL The amplitde A can be any complex number with |A| sqrtfracL
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Exercise:
The eigenstates for a particle in an infinite square potential well are given by psi_nxt Asink_n x e^-iomega_n t Show that for the normalised wave functions i.e. for _^L|psi_nxt|^textrmdx the amplitude has to be A sqrtfracL

Solution:
_^L|psi_nxt|^textrmdx _^L|Asink_n x e^-iomega_n t|^ textrmdx _^L|A|^sin^k_n x |e^-iomega_n t|^ textrmdx |A|^ _^L sin^k_n xtextrmdx where we use the fact that e^iphi is a unit vector in the complex plane. vspacemm Using the identiy sin^phi fracleft-cos phiright we can write the egral as dots frac|A|^ _^L left-cos k_n xright textrmdx frac|A|^ left _^L textrmdx -_^L cos k_n x textrmdx right frac|A|^ L-frac|A|^ k_n sin k_n xBig|_^L frac|A|^L-frac|A|^ k_nsink_n xBig|_^L The sine function has zeros at and also at L by definition of k_n so the second term vanishes. The normalisation condition can thus be written as frac|A|^L Longrightarrow |A|^ fracL The amplitde A can be any complex number with |A| sqrtfracL
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quantum physics
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normalisation, potential well, schrödinger equation
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