  We study the quantum dynamics of a Bose Josephson junction(BJJ) made up of
two coupled Bose-Einstein condensates. Apart from the usual ac Josephson
oscillations, two different dynamical states of BJJ can be observed by tuning
the inter-particle interaction strength, which are known as '$\pi$-oscillation'
with relative phase $\pi$ between the condensates and 'macroscopic
self-trapped' (MST) state with finite number imbalance. By choosing appropiate
intial state we study above dynamical branches quantum mechanically and compare
with classical dynamics. The stability region of the '$\pi$-oscillation' is
separated from that of 'MST' state at a critical coupling strength. Also a
significant change in the energy spectrum takes place above the critical
coupling strength, and pairs of (quasi)-degenerate excited states appear. The
original model of BJJ can be mapped on to a simple Hamiltonian describing
quantum particle in an 'effective potential' with an effective Planck constant.
Different dynamical states and degenerate excited states in the energy spectrum
can be understood in this 'effective potential' approach. Also possible novel
quantum phenomena like 'macroscopic quantum tunneling'(MQT) become evident from
the simple picture of 'effective potential'. We study decay of metastable
'$\pi$-oscillation' by MQT through potential barrier. The doubly degenerate
excited states in the energy spectrum are associated with the classically
degenerate MST states with equal and opposite number imbalance. We calculate
the energy splitting between these quasi-degenerate excited states due to MQT
of the condensate between classically degenerate MST states.
