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Chemistry LibreTexts

419: The Variation Theorem in Dirac Notation

  • Page ID
    135909
  • The recipe for calculating the expectation value for energy using a trial wave function is,

    \[ \langle E \rangle = \langle \psi | \hat{H} | \psi \rangle \label{1}\]

    Now suppose the eigenvalues of \( \hat{H}\) are denoted by \( |i \rangle\). Then,

    \[ \hat{H} |i \rangle = \varepsilon_{i} |i \rangle = |i \rangle \varepsilon _{i} \label{2}\]

    Next we write \( | \psi \rangle\) as a superposition of the eigenfunctions \( |i \rangle\),

    \[ | \psi \rangle = \sum_{i} |i \rangle \langle I| \psi \rangle\]

    and substitute it into Equation \ref{1}.

    \[ \langle E \rangle = \sum_{i} \langle \psi | \hat{H} | i \rangle \langle i | \psi \rangle\]

    Making use of Equation \ref{2} yields,

    \[ \langle E \rangle = \sum_{i} \langle \psi |i \rangle \varepsilon_{i} \langle i | \psi \rangle\]

    After rearrangement we have,

    \[ \langle E \rangle = \sum_{i} \varepsilon_{i} | \langle i| \psi \rangle |^{2}\]

    However, \( | \langle i | \psi \rangle |^{2}\) is the probability that \( \varepsilon_{i}\) will be observed, \(p_i\).

    \[ \langle E \rangle = \sum_{i} \varepsilon_{i} p_{i} \geq \varepsilon_{0}\]

    Thus, the expectation value obtained using the trial wave function is an upper bound to the true energy. In other words, in valid quantum mechanical calculations you can't get a lower energy than the true energy.