This is a detailed version of calculations originally done in Machine Learning Catalysis of quantum tunneling. Thank you to Emmanuele Cinnirella for sharing the calculations below


We have an Hamiltonian given by , where , and:


Given the initial state , where and , we want to calculate the Quantum tunnelling probability .


The value of interest is . Since we can ignore all terms in that do not involve . This means we only care about the term, getting the result

We maximise the Quantum tunnelling probability when = and, for the given , we have