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The analytical transfer matrix technique is applied to the Schr\"{o}dingerequation of symmetric quartic-well potential problem in the form$V(x)={1/2}kx^{2}+\lambda{x^{4}}.$ This gives quantization condition from whichwe can calculate the ground-state energy eigenvalues numerically. We alsocompare the results with those obtained from numerical shooting method,perturbation theory, and WKB method.

Ground-state energy eigenvalue calculation of the quantum mechanical well $V(x)={1/2}kx^{2}+\lambda {x^{4}}$ via analytical transfer matrix method