Editing Bound state (section)

=== Non-degeneracy in one-dimensional bound states ===
One-dimensional bound states can be shown to be non-degenerate in energy for well-behaved wavefunctions that decay to zero at infinities. This need not hold true for wavefunctions in higher dimensions. Due to the property of non-degenerate states, one-dimensional bound states can always be expressed as real wavefunctions.

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!Proof
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Consider two energy eigenstates states <math display="inline"> \Psi_1</math> and <math display="inline"> \Psi_2</math> with same energy eigenvalue.

Then since, the Schrodinger equation, which is expressed as:<math display="block">E  = - \frac 1 {\Psi_i(x,t)} \frac{\hbar^2}{2m}\frac{\partial^2\Psi_i(x,t) }{\partial x^2} + V(x,t) </math>is satisfied for i = 1 and 2, subtracting the two equations gives:<math display="block">\frac 1 {\Psi_1(x,t)} \frac{\partial^2\Psi_1(x,t) }{\partial x^2} - \frac 1 {\Psi_2(x,t)} \frac{\partial^2\Psi_2(x,t) }{\partial x^2} = 0     </math>which can be rearranged to give the condition:<math display="block">
\frac{\partial }{\partial x} \left(\frac{\partial \Psi_1}{\partial x}\Psi_2\right)-\frac{\partial }{\partial x} \left(\frac{\partial \Psi_2}{\partial x}\Psi_1\right)=0  </math>Since <math display="inline">
\frac{\partial \Psi_1}{\partial x}(x)\Psi_2(x)- 
\frac{\partial \Psi_2}{\partial x}(x)\Psi_1(x)= C  </math>, taking limit of x going to infinity on both sides, the wavefunctions vanish and gives <math display="inline">
C = 0  </math>.


Solving for <math display="inline">
\frac{\partial \Psi_1}{\partial x}(x)\Psi_2(x) =
\frac{\partial \Psi_2}{\partial x}(x)\Psi_1(x) </math>, we get: <math display="inline">
\Psi_1(x) = k \Psi_2(x)       </math> which proves that the energy eigenfunction of a 1D bound state is unique.



Furthermore it can be shown that these wavefunctions can always be represented by a completely real wavefunction. Define real functions <math display="inline">
\rho_1(x) </math> and <math display="inline">
\rho_2(x)   </math> such that <math display="inline">
\Psi(x) = \rho_1(x) + i \rho_2(x) </math>. Then, from Schrodinger's equation:<math display="block">\Psi'' = - \frac{2m(E-V(x))}{\hbar^2}\Psi </math> we get that, since the terms in the equation are all real values:<math display="block">\rho_i'' = - \frac{2m(E-V(x))}{\hbar^2}\rho_i   </math>applies for i = 1 and 2. Thus every 1D bound state can be represented by completely real eigenfunctions. Note that real function representation of wavefunctions from this proof applies for all non-degenerate states in general. 
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